text: r"""
  Free modules
  
  Sage supports computation with free modules over an arbitrary commutative ring.
  Nontrivial functionality is available over `\ZZ`, fields, and some principal
  ideal domains (e.g. `\QQ[x]` and rings of integers of number fields). All free
  modules over an integral domain are equipped with an embedding in an ambient
  vector space and an inner product, which you can specify and change.
  
  Create the free module of rank `n` over an arbitrary commutative ring `R` using
  the command ``FreeModule(R,n)``. Equivalently, ``R^n`` also creates that free
  module.
  
  The following example illustrates the creation of both a vector space and a
  free module over the integers and a submodule of it.  Use the functions
  ``FreeModule``, ``span`` and member functions of free modules to create free
  modules.  *Do not use the FreeModule_xxx constructors directly.*
  
  EXAMPLES::
  
      sage: V = VectorSpace(QQ,3)
      sage: W = V.subspace([[1,2,7], [1,1,0]])
      sage: W
      Vector space of degree 3 and dimension 2 over Rational Field
      Basis matrix:
      [ 1  0 -7]
      [ 0  1  7]
      sage: C = VectorSpaces(FiniteField(7))
      sage: C
      Category of vector spaces over Finite Field of size 7
      sage: C(W)
      Vector space of degree 3 and dimension 2 over Finite Field of size 7
      Basis matrix:
      [1 0 0]
      [0 1 0]
  
  ::
  
      sage: M = ZZ^3
      sage: C = VectorSpaces(FiniteField(7))
      sage: C(M)
      Vector space of dimension 3 over Finite Field of size 7
      sage: W = M.submodule([[1,2,7], [8,8,0]])
      sage: C(W)
      Vector space of degree 3 and dimension 2 over Finite Field of size 7
      Basis matrix:
      [1 0 0]
      [0 1 0]
  
  We illustrate the exponent notation for creation of free modules.
  
  ::
  
      sage: ZZ^4
      Ambient free module of rank 4 over the principal ideal domain Integer Ring
      sage: QQ^2
      Vector space of dimension 2 over Rational Field
      sage: RR^3
      Vector space of dimension 3 over Real Field with 53 bits of precision
  
  Base ring::
  
      sage: R.<x,y> = QQ[]
      sage: M = FreeModule(R,2)
      sage: M.base_ring()
      Multivariate Polynomial Ring in x, y over Rational Field
  
  ::
  
      sage: VectorSpace(QQ, 10).base_ring()
      Rational Field
  
  TESTS:
  
  We intersect a zero-dimensional vector space with a
  1-dimension submodule.
  
  ::
  
      sage: V = (QQ^1).span([])
      sage: W = ZZ^1
      sage: V.intersection(W)
      Free module of degree 1 and rank 0 over Integer Ring
      Echelon basis matrix:
      []
  
  We construct subspaces of real and complex double vector spaces and
  verify that the element types are correct::
  
      sage: V = FreeModule(RDF, 3); V
      Vector space of dimension 3 over Real Double Field
      sage: V.0
      (1.0, 0.0, 0.0)
      sage: type(V.0)
      <class 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
      sage: W = V.span([V.0]); W
      Vector space of degree 3 and dimension 1 over Real Double Field
      Basis matrix:
      [1.0 0.0 0.0]
      sage: type(W.0)
      <class 'sage.modules.vector_real_double_dense.Vector_real_double_dense'>
      sage: V = FreeModule(CDF, 3); V
      Vector space of dimension 3 over Complex Double Field
      sage: type(V.0)
      <class 'sage.modules.vector_complex_double_dense.Vector_complex_double_dense'>
      sage: W = V.span_of_basis([CDF.0 * V.1]); W
      Vector space of degree 3 and dimension 1 over Complex Double Field
      User basis matrix:
      [  0.0 1.0*I   0.0]
      sage: type(W.0)
      <class 'sage.modules.vector_complex_double_dense.Vector_complex_double_dense'>
  
  Basis vectors are immutable::
  
      sage: A = span([[1,2,3], [4,5,6]], ZZ)
      sage: A.0
      (1, 2, 3)
      sage: A.0[0] = 5
      Traceback (most recent call last):
      ...
      ValueError: vector is immutable; please change a copy instead (use copy())
  
  Among other things, this tests that we can save and load submodules
  and elements::
  
      sage: M = ZZ^3
      sage: TestSuite(M).run()
      sage: W = M.span_of_basis([[1,2,3],[4,5,19]])
      sage: TestSuite(W).run()
      sage: v = W.0 + W.1
      sage: TestSuite(v).run()
  
  AUTHORS:
  
  - William Stein (2005, 2007)
  
  - David Kohel (2007, 2008)
  
  - Niles Johnson (2010-08): (:trac:`3893`) ``random_element()`` should pass on ``*args`` and ``**kwds``.
  
  - Simon King (2010-12):
    :trac:`8800` : Fixing a bug in ``denominator()``.
  
  - Simon King (2010-12), Peter Bruin (June 2014):
    :trac:`10513` : New coercion model and category framework.
  
  """
  
  ###########################################################################
  #       Copyright (C) 2005, 2007 William Stein <wstein@gmail.com>
  #       Copyright (C) 2007, 2008 David Kohel <kohel@iml.univ-mrs.fr>
  #
  #  Distributed under the terms of the GNU General Public License (GPL)
  #
  #    This code is distributed in the hope that it will be useful,
  #    but WITHOUT ANY WARRANTY; without even the implied warranty of
  #    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  #    General Public License for more details.
  #
  #  The full text of the GPL is available at:
  #
  #                  http://www.gnu.org/licenses/
  ###########################################################################
  from itertools import islice
  
  from . import free_module_element
  import sage.matrix.matrix_space
  import sage.misc.latex as latex
  
  from sage.modules.module import Module
  import sage.rings.finite_rings.finite_field_constructor as finite_field
  import sage.rings.ring as ring
  import sage.rings.integer_ring
  import sage.rings.rational_field
  import sage.rings.finite_rings.integer_mod_ring
  import sage.rings.infinity
  import sage.rings.integer
  from sage.categories.principal_ideal_domains import PrincipalIdealDomains
  from sage.misc.randstate import current_randstate
  from sage.structure.sequence import Sequence
  from sage.structure.richcmp import (richcmp_method, rich_to_bool, richcmp,
                                      richcmp_not_equal, revop,
                                      op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE)
  from sage.misc.cachefunc import cached_method
  
  from warnings import warn
  
  ###############################################################################
  #
  # Constructor functions
  #
  ###############################################################################
  
  from sage.structure.factory import UniqueFactory
  
  class FreeModuleFactory(UniqueFactory):
      r"""
      Factory class for the finite-dimensional free modules with standard basis
      """
      def create_key(self, base_ring, rank, sparse=False, inner_product_matrix=None):
          """
          TESTS::
  
              sage: loads(dumps(ZZ^6)) is ZZ^6
              True
              sage: loads(dumps(RDF^3)) is RDF^3
              True
  
          .. TODO::
  
              Replace the above by ``TestSuite(...).run()``, once
              :meth:`_test_pickling` will test unique representation
              and not only equality.
          """
          rank = int(sage.rings.integer.Integer(rank))
  
          if not (inner_product_matrix is None):
              inner_product_matrix = sage.matrix.matrix_space.MatrixSpace(base_ring, rank)(inner_product_matrix)
              inner_product_matrix.set_immutable()
  
          return (base_ring, rank, sparse, inner_product_matrix)
  
      def create_object(self, version, key):
  
          base_ring, rank, sparse, inner_product_matrix = key
  
          if inner_product_matrix is not None:
              from .free_quadratic_module import FreeQuadraticModule
              return FreeQuadraticModule(base_ring, rank, inner_product_matrix=inner_product_matrix, sparse=sparse)
  
          if not isinstance(sparse,bool):
              raise TypeError("Argument sparse (= %s) must be True or False" % sparse)
  
          if not (hasattr(base_ring,'is_commutative') and base_ring.is_commutative()):
              warn("""You are constructing a free module
  over a noncommutative ring. Sage does not have a concept
  of left/right and both sided modules, so be careful.
  It's also not guaranteed that all multiplications are
  done from the right side.""")
  
          #            raise TypeError, "The base_ring must be a commutative ring."
  
          try:
              if not sparse and isinstance(base_ring,sage.rings.real_double.RealDoubleField_class):
                  return RealDoubleVectorSpace_class(rank)
  
              elif not sparse and isinstance(base_ring,sage.rings.complex_double.ComplexDoubleField_class):
                  return ComplexDoubleVectorSpace_class(rank)
  
              elif base_ring.is_field():
                  return FreeModule_ambient_field(base_ring, rank, sparse=sparse)
  
              elif base_ring in PrincipalIdealDomains():
                  return FreeModule_ambient_pid(base_ring, rank, sparse=sparse)
  
              elif isinstance(base_ring, sage.rings.number_field.order.Order) \
                  and base_ring.is_maximal() and base_ring.class_number() == 1:
                  return FreeModule_ambient_pid(base_ring, rank, sparse=sparse)
  
              elif isinstance(base_ring, ring.IntegralDomain) or base_ring.is_integral_domain():
                  return FreeModule_ambient_domain(base_ring, rank, sparse=sparse)
  
              else:
                  return FreeModule_ambient(base_ring, rank, sparse=sparse)
          except NotImplementedError:
              return FreeModule_ambient(base_ring, rank, sparse=sparse)
  
  FreeModuleFactory_with_standard_basis = FreeModuleFactory("FreeModule")
  
  def FreeModule(base_ring, rank_or_basis_keys=None, sparse=False, inner_product_matrix=None, *,
                 with_basis='standard', rank=None, basis_keys=None, **args):
      r"""
      Create a free module over the given commutative ``base_ring``
  
      ``FreeModule`` can be called with the following positional arguments:
  
      - ``FreeModule(base_ring, rank, ...)``
  
      - ``FreeModule(base_ring, basis_keys, ...)``
  
      INPUT:
  
      -  ``base_ring`` - a commutative ring
  
      -  ``rank`` - a nonnegative integer
  
      -  ``basis_keys`` - a finite or enumerated family of arbitrary objects
  
      -  ``sparse`` - bool; (default False)
  
      -  ``inner_product_matrix`` - the inner product matrix (default ``None``)
  
      -  ``with_basis`` - either ``"standard"`` (the default), in which case
         a free module with the standard basis as the distinguished basis is created;
         or ``None``, in which case a free module without distinguished basis is
         created.
  
      -  further options may be accepted by various implementation classes
  
      OUTPUT: a free module
  
      This factory function creates instances of various specialized classes
      depending on the input.  Not all combinations of options are
      implemented.
  
      -  If the parameter ``basis_keys`` is provided, it must be a finite
         or enumerated family of objects, and an instance of
         :class:`CombinatorialFreeModule` is created.
  
         EXAMPLES::
  
             sage: CombinatorialFreeModule(QQ, ['a','b','c'])
             Free module generated by {'a', 'b', 'c'} over Rational Field
  
         It has a distinguished standard basis that is indexed by the provided
         ``basis_keys``. See the documentation of :class:`CombinatorialFreeModule`
         for more examples and details, including its :class:`UniqueRepresentation`
         semantics.
  
      -  If the parameter ``with_basis`` is set to ``None``, then a free module
         of the given ``rank`` without distinguished basis is created.  It is
         represented by an instance of :class:`FiniteRankFreeModule`.
  
         EXAMPLES::
  
             sage: FiniteRankFreeModule(ZZ, 3, name='M')
             Rank-3 free module M over the Integer Ring
  
         See the documentation of :class:`FiniteRankFreeModule` for more
         options, examples, and details.
  
      -  If ``rank`` is provided and the option ``with_basis`` is left at its
         default value, ``"standard"``, then a free ambient module with
         distinguished standard basis indexed by ``range(rank)`` is created.
         There is only one dense and one sparse free ambient module of
         given ``rank`` over ``base_ring``.
  
         EXAMPLES::
  
             sage: FreeModule(Integers(8), 10)
             Ambient free module of rank 10 over Ring of integers modulo 8
  
         The remainder of this documentation discusses this case of
         free ambient modules.
  
  
      EXAMPLES:
  
      First we illustrate creating free modules over various base fields.
      The base field affects the free module that is created. For
      example, free modules over a field are vector spaces, and free
      modules over a principal ideal domain are special in that more
      functionality is available for them than for completely general
      free modules.
  
      ::
  
          sage: FreeModule(QQ,10)
          Vector space of dimension 10 over Rational Field
          sage: FreeModule(ZZ,10)
          Ambient free module of rank 10 over the principal ideal domain Integer Ring
          sage: FreeModule(FiniteField(5),10)
          Vector space of dimension 10 over Finite Field of size 5
          sage: FreeModule(Integers(7),10)
          Vector space of dimension 10 over Ring of integers modulo 7
          sage: FreeModule(PolynomialRing(QQ,'x'),5)
          Ambient free module of rank 5 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
          sage: FreeModule(PolynomialRing(ZZ,'x'),5)
          Ambient free module of rank 5 over the integral domain Univariate Polynomial Ring in x over Integer Ring
  
      Of course we can make rank 0 free modules::
  
          sage: FreeModule(RealField(100),0)
          Vector space of dimension 0 over Real Field with 100 bits of precision
  
      Next we create a free module with sparse representation of
      elements. Functionality with sparse modules is *identical* to dense
      modules, but they may use less memory and arithmetic may be faster
      (or slower!).
  
      ::
  
          sage: M = FreeModule(ZZ,200,sparse=True)
          sage: M.is_sparse()
          True
          sage: type(M.0)
          <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
  
      The default is dense.
  
      ::
  
          sage: M = ZZ^200
          sage: type(M.0)
          <class 'sage.modules.vector_integer_dense.Vector_integer_dense'>
  
      Note that matrices associated in some way to sparse free modules
      are sparse by default::
  
          sage: M = FreeModule(Integers(8), 2)
          sage: A = M.basis_matrix()
          sage: A.is_sparse()
          False
          sage: Ms = FreeModule(Integers(8), 2, sparse=True)
          sage: M == Ms  # as mathematical objects they are equal
          True
          sage: Ms.basis_matrix().is_sparse()
          True
  
      We can also specify an inner product matrix, which is used when
      computing inner products of elements.
  
      ::
  
          sage: A = MatrixSpace(ZZ,2)([[1,0],[0,-1]])
          sage: M = FreeModule(ZZ,2,inner_product_matrix=A)
          sage: v, w = M.gens()
          sage: v.inner_product(w)
          0
          sage: v.inner_product(v)
          1
          sage: w.inner_product(w)
          -1
          sage: (v+2*w).inner_product(w)
          -2
  
      You can also specify the inner product matrix by giving anything
      that coerces to an appropriate matrix. This is only useful if the
      inner product matrix takes values in the base ring.
  
      ::
  
          sage: FreeModule(ZZ,2,inner_product_matrix=1).inner_product_matrix()
          [1 0]
          [0 1]
          sage: FreeModule(ZZ,2,inner_product_matrix=[1,2,3,4]).inner_product_matrix()
          [1 2]
          [3 4]
          sage: FreeModule(ZZ,2,inner_product_matrix=[[1,2],[3,4]]).inner_product_matrix()
          [1 2]
          [3 4]
  
      .. todo::
  
          Refactor modules such that it only counts what category the base
          ring belongs to, but not what is its Python class.
  
  
      EXAMPLES::
  
          sage: FreeModule(QQ, ['a', 'b', 'c'])
          Free module generated by {'a', 'b', 'c'} over Rational Field
          sage: _.category()
          Category of finite dimensional vector spaces with basis over Rational Field
  
          sage: FreeModule(QQ, 3, with_basis=None)
          3-dimensional vector space over the Rational Field
          sage: _.category()
          Category of finite dimensional vector spaces over Rational Field
  
      """
      if rank_or_basis_keys is not None:
          try:
              rank = sage.rings.integer_ring.ZZ(rank_or_basis_keys)
          except:
              basis_keys = rank_or_basis_keys
      if not with_basis:
          if inner_product_matrix is not None:
              raise NotImplementedError
          from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule
          return FiniteRankFreeModule(base_ring, rank, **args)
      elif with_basis == 'standard':
          if rank is not None:
              return FreeModuleFactory_with_standard_basis(base_ring, rank, sparse,
                                                          inner_product_matrix, **args)
          else:
              if inner_product_matrix is not None:
                  raise NotImplementedError
              from sage.combinat.free_module import CombinatorialFreeModule
              return CombinatorialFreeModule(base_ring, basis_keys, **args)
      else:
          raise NotImplementedError
  
  def VectorSpace(K, dimension_or_basis_keys=None, sparse=False, inner_product_matrix=None, *,
                  with_basis='standard', dimension=None, basis_keys=None, **args):
      """
      EXAMPLES:
  
      The base can be complicated, as long as it is a field.
  
      ::
  
          sage: V = VectorSpace(FractionField(PolynomialRing(ZZ,'x')),3)
          sage: V
          Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in x over Integer Ring
          sage: V.basis()
          [
          (1, 0, 0),
          (0, 1, 0),
          (0, 0, 1)
          ]
  
      The base must be a field or a ``TypeError`` is raised.
  
      ::
  
          sage: VectorSpace(ZZ,5)
          Traceback (most recent call last):
          ...
          TypeError: Argument K (= Integer Ring) must be a field.
      """
      if not K.is_field():
          raise TypeError("Argument K (= %s) must be a field." % K)
      if not sparse in (True,False):
          raise TypeError("Argument sparse (= %s) must be a boolean."%sparse)
      return FreeModule(K, dimension_or_basis_keys, sparse, inner_product_matrix,
                        with_basis=with_basis, rank=dimension, basis_keys=basis_keys,
                        **args)
  
  ###############################################################################
  #
  # The span of vectors
  #
  ###############################################################################
  
  def span(gens, base_ring=None, check=True, already_echelonized=False):
      r"""
      Return the span of the vectors in ``gens`` using scalars from ``base_ring``.
  
      INPUT:
  
      - ``gens`` - a list of either vectors or lists of ring elements
        used to generate the span
  
      - ``base_ring`` - default: ``None`` - a principal ideal domain
        for the ring of scalars
  
      - ``check`` - default: ``True`` - passed to the ``span()`` method
        of the ambient module
  
      - ``already_echelonized`` - default: ``False`` - set to ``True``
        if the vectors form the rows of a matrix in echelon form, in
        order to skip the computation of an echelonized basis for the
        span.
  
      OUTPUT:
  
      A module (or vector space) that is all the linear combinations of the
      free module elements (or vectors) with scalars from the
      ring (or field) given by ``base_ring``.  See the examples below
      describing behavior when the base ring is not specified and/or
      the module elements are given as lists that do not carry
      explicit base ring information.
  
      EXAMPLES:
  
      The vectors in the list of generators can be given as
      lists, provided a base ring is specified and the elements of the list
      are in the ring (or the fraction field of the ring).  If the
      base ring is a field, the span is a vector space.  ::
  
          sage: V = span([[1,2,5], [2,2,2]], QQ); V
          Vector space of degree 3 and dimension 2 over Rational Field
          Basis matrix:
          [ 1  0 -3]
          [ 0  1  4]
  
          sage: span([V.gen(0)], QuadraticField(-7,'a'))
          Vector space of degree 3 and dimension 1 over Number Field in a with defining polynomial x^2 + 7 with a = 2.645751311064591?*I
          Basis matrix:
          [ 1  0 -3]
  
          sage: span([[1,2,3], [2,2,2], [1,2,5]], GF(2))
          Vector space of degree 3 and dimension 1 over Finite Field of size 2
          Basis matrix:
          [1 0 1]
  
      If the base ring is not a field, then a module is created.
      The entries of the vectors can lie outside the ring, if they
      are in the fraction field of the ring.  ::
  
          sage: span([[1,2,5], [2,2,2]], ZZ)
          Free module of degree 3 and rank 2 over Integer Ring
          Echelon basis matrix:
          [ 1  0 -3]
          [ 0  2  8]
  
          sage: span([[1,1,1], [1,1/2,1]], ZZ)
          Free module of degree 3 and rank 2 over Integer Ring
          Echelon basis matrix:
          [  1   0   1]
          [  0 1/2   0]
  
          sage: R.<x> = QQ[]
          sage: M= span( [[x, x^2+1], [1/x, x^3]], R); M
          Free module of degree 2 and rank 2 over
          Univariate Polynomial Ring in x over Rational Field
          Echelon basis matrix:
          [          1/x           x^3]
          [            0 x^5 - x^2 - 1]
          sage: M.basis()[0][0].parent()
          Fraction Field of Univariate Polynomial Ring in x over Rational Field
  
      A base ring can be inferred if the generators are given as a
      list of vectors. ::
  
          sage: span([vector(QQ, [1,2,3]), vector(QQ, [4,5,6])])
          Vector space of degree 3 and dimension 2 over Rational Field
          Basis matrix:
          [ 1  0 -1]
          [ 0  1  2]
          sage: span([vector(QQ, [1,2,3]), vector(ZZ, [4,5,6])])
          Vector space of degree 3 and dimension 2 over Rational Field
          Basis matrix:
          [ 1  0 -1]
          [ 0  1  2]
          sage: span([vector(ZZ, [1,2,3]), vector(ZZ, [4,5,6])])
          Free module of degree 3 and rank 2 over Integer Ring
          Echelon basis matrix:
          [1 2 3]
          [0 3 6]
  
      TESTS::
  
          sage: span([[1,2,3], [2,2,2], [1,2/3,5]], ZZ)
          Free module of degree 3 and rank 3 over Integer Ring
          Echelon basis matrix:
          [  1   0  13]
          [  0 2/3   6]
          [  0   0  14]
          sage: span([[1,2,3], [2,2,2], [1,2,QQ['x'].gen()]], ZZ)
          Traceback (most recent call last):
          ...
          ValueError: The elements of gens (= [[1, 2, 3], [2, 2, 2], [1, 2, x]]) must be defined over base_ring (= Integer Ring) or its field of fractions.
  
      For backwards compatibility one can also give the base ring as the
      first argument.  ::
  
          sage: span(QQ,[[1,2],[3,4]])
          Vector space of degree 2 and dimension 2 over Rational Field
          Basis matrix:
          [1 0]
          [0 1]
  
      The base ring must be a principal ideal domain (PID).  ::
  
          sage: span([[1,2,3]], Integers(6))
          Traceback (most recent call last):
          ...
          TypeError: The base_ring (= Ring of integers modulo 6)
          must be a principal ideal domain.
  
      Fix :trac:`5575`::
  
          sage: V = QQ^3
          sage: span([V.0, V.1])
          Vector space of degree 3 and dimension 2 over Rational Field
          Basis matrix:
          [1 0 0]
          [0 1 0]
  
      Improve error message from :trac:`12541`::
  
          sage: span({0:vector([0,1])}, QQ)
          Traceback (most recent call last):
          ...
          TypeError: generators must be lists of ring elements
          or free module elements!
      """
      if ring.is_Ring(gens):
          # we allow the old input format with first input the base_ring.
          # Do we want to deprecate it?..
          base_ring, gens = gens, base_ring
  
      try:
          if base_ring is None:
              gens = Sequence(gens)
              R = gens.universe().base_ring()
          else:
              gens = list(gens)
              R = base_ring
      except TypeError:
          raise TypeError("generators must be given as an iterable structure!")
  
      if R not in PrincipalIdealDomains():
          raise TypeError("The base_ring (= %s) must be a principal ideal "
                          "domain." % R)
      if not gens:
          return FreeModule(R, 0)
      else:
          x = gens[0]
          if free_module_element.is_FreeModuleElement(x):
              M = x.parent()
          else:
              try:
                  x = list(x)
              except TypeError:
                  raise TypeError("generators must be lists of ring elements or "
                                  "free module elements!")
              M = FreeModule(R, len(x))
              try:
                  gens = [M(_) for _ in gens]
              except TypeError:
                  R = R.fraction_field()
                  M = FreeModule(R, len(x))
                  try:
                      gens = [M(_) for _ in gens]
                  except TypeError:
                      raise ValueError("The elements of gens (= %s) must be "
                                       "defined over base_ring (= %s) or its "
                                       "field of fractions." % (gens, base_ring))
          return M.span(gens=gens, base_ring=base_ring, check=check,
                        already_echelonized=already_echelonized)
  
  ###############################################################################
  #
  # Base class for all free modules
  #
  ###############################################################################
  
  def is_FreeModule(M):
      """
      Return True if M inherits from FreeModule_generic.
  
      EXAMPLES::
  
          sage: from sage.modules.free_module import is_FreeModule
          sage: V = ZZ^3
          sage: is_FreeModule(V)
          True
          sage: W = V.span([ V.random_element() for i in range(2) ])
          sage: is_FreeModule(W)
          True
      """
      return isinstance(M, FreeModule_generic)
  
  @richcmp_method
  class FreeModule_generic(Module):
      """
      Base class for all free modules.
  
      TESTS:
  
      Check that :trac:`17576` is fixed::
  
          sage: V = VectorSpace(RDF, 3)
          sage: v = vector(RDF, [1, 2, 3, 4])
          sage: v in V
          False
      """
      def __init__(self, base_ring, rank, degree, sparse=False,
                   coordinate_ring=None, category=None):
          """
          Create the free module of given rank ``rank`` over the given base
          ring ``base_ring``.
  
          INPUT:
  
          - ``base_ring`` -- a commutative ring
  
          - ``rank`` -- a non-negative integer
  
          - ``degree`` -- a non-negative integer
  
          - ``sparse`` -- bool (default: False)
  
          - ``coordinate_ring`` -- a ring containing ``base_ring``
            (default: equal to ``base_ring``)
  
          - ``category`` -- category (default: None)
  
          If ``base_ring`` is a field, then the default category is the
          category of finite-dimensional vector spaces over that field;
          otherwise it is the category of finite-dimensional free modules
          over that ring.  In addition, the category is intersected with the
          category of finite enumerated sets if the ring is finite or the
          rank is 0.
  
          EXAMPLES::
  
              sage: PolynomialRing(QQ,3,'x')^3
              Ambient free module of rank 3 over the integral domain Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
  
              sage: FreeModule(GF(7),3).category()
              Category of enumerated finite dimensional vector spaces with basis over
               (finite enumerated fields and subquotients of monoids and quotients of semigroups)
              sage: V = QQ^4; V.category()
              Category of finite dimensional vector spaces with basis over
               (number fields and quotient fields and metric spaces)
              sage: V = GF(5)**20; V.category()
              Category of enumerated finite dimensional vector spaces with basis over (finite enumerated fields and subquotients of monoids and quotients of semigroups)
              sage: FreeModule(ZZ,3).category()
              Category of finite dimensional modules with basis over
               (euclidean domains and infinite enumerated sets
                and metric spaces)
              sage: (QQ^0).category()
              Category of finite enumerated finite dimensional vector spaces with basis
               over (number fields and quotient fields and metric spaces)
  
          TESTS::
  
              sage: M = FreeModule(ZZ,20,sparse=False)
              sage: x = M.random_element()
              sage: type(x)
              <class 'sage.modules.vector_integer_dense.Vector_integer_dense'>
              sage: M.element_class
              <class 'sage.modules.vector_integer_dense.Vector_integer_dense'>
  
              sage: N = FreeModule(ZZ,20,sparse=True)
              sage: y = N.random_element()
              sage: type(y)
              <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
              sage: N.element_class
              <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
  
          """
          if not base_ring.is_commutative():
              warn("""You are constructing a free module
  over a noncommutative ring. Sage does not have a concept
  of left/right and both sided modules, so be careful.
  It's also not guaranteed that all multiplications are
  done from the right side.""")
  
          if coordinate_ring is None:
              coordinate_ring = base_ring
  
          if not hasattr(self, 'Element'):
              self.Element = element_class(coordinate_ring, sparse)
  
          rank = sage.rings.integer.Integer(rank)
          if rank < 0:
              raise ValueError("rank (=%s) must be nonnegative"%rank)
          degree = sage.rings.integer.Integer(degree)
          if degree < 0:
              raise ValueError("degree (=%s) must be nonnegative"%degree)
  
          if category is None:
              from sage.categories.all import FreeModules
              category = FreeModules(base_ring.category()).FiniteDimensional()
              try:
                  if base_ring.is_finite() or rank == 0:
                      category = category.Enumerated().Finite()
              except Exception:
                  pass
  
          super(FreeModule_generic, self).__init__(base_ring, category=category)
          self.__coordinate_ring = coordinate_ring
          self.__uses_ambient_inner_product = True
          self.__rank = rank
          self.__degree = degree
          self.__is_sparse = sparse
          self._gram_matrix = None
  
      def construction(self):
          """
          The construction functor and base ring for self.
  
          EXAMPLES::
  
              sage: R = PolynomialRing(QQ,3,'x')
              sage: V = R^5
              sage: V.construction()
              (VectorFunctor, Multivariate Polynomial Ring in x0, x1, x2 over Rational Field)
          """
          from sage.categories.pushout import VectorFunctor
          if hasattr(self,'_inner_product_matrix'):
              return VectorFunctor(self.rank(), self.is_sparse(),self.inner_product_matrix()), self.base_ring()
          return VectorFunctor(self.rank(), self.is_sparse()), self.base_ring()
  
      # FIXME: what's the level of generality of FreeModuleHomspace?
      # Should there be a category for free modules accepting it as hom space?
      # See similar method for FreeModule_generic_field class
      def _Hom_(self, Y, category):
          from .free_module_homspace import FreeModuleHomspace
          return FreeModuleHomspace(self, Y, category)
  
      def dense_module(self):
          """
          Return corresponding dense module.
  
          EXAMPLES:
  
          We first illustrate conversion with ambient spaces::
  
              sage: M = FreeModule(QQ,3)
              sage: S = FreeModule(QQ,3, sparse=True)
              sage: M.sparse_module()
              Sparse vector space of dimension 3 over Rational Field
              sage: S.dense_module()
              Vector space of dimension 3 over Rational Field
              sage: M.sparse_module() == S
              True
              sage: S.dense_module() == M
              True
              sage: M.dense_module() == M
              True
              sage: S.sparse_module() == S
              True
  
          Next we create a subspace::
  
              sage: M = FreeModule(QQ,3, sparse=True)
              sage: V = M.span([ [1,2,3] ] ); V
              Sparse vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [1 2 3]
              sage: V.sparse_module()
              Sparse vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [1 2 3]
          """
          if self.is_sparse():
              return self._dense_module()
          return self
  
      def _dense_module(self):
          """
          Creates a dense module with the same defining data as self.
  
          N.B. This function is for internal use only! See dense_module for
          use.
  
          EXAMPLES::
  
              sage: M = FreeModule(Integers(8),3)
              sage: S = FreeModule(Integers(8),3, sparse=True)
              sage: M is S._dense_module()
              True
          """
          A = self.ambient_module().dense_module()
          return A.span(self.basis())
  
      def sparse_module(self):
          """
          Return the corresponding sparse module with the same defining
          data.
  
          EXAMPLES:
  
          We first illustrate conversion with ambient spaces::
  
              sage: M = FreeModule(Integers(8),3)
              sage: S = FreeModule(Integers(8),3, sparse=True)
              sage: M.sparse_module()
              Ambient sparse free module of rank 3 over Ring of integers modulo 8
              sage: S.dense_module()
              Ambient free module of rank 3 over Ring of integers modulo 8
              sage: M.sparse_module() is S
              True
              sage: S.dense_module() is M
              True
              sage: M.dense_module() is M
              True
              sage: S.sparse_module() is S
              True
  
          Next we convert a subspace::
  
              sage: M = FreeModule(QQ,3)
              sage: V = M.span([ [1,2,3] ] ); V
              Vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [1 2 3]
              sage: V.sparse_module()
              Sparse vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [1 2 3]
          """
          if self.is_sparse():
              return self
          return self._sparse_module()
  
      def _sparse_module(self):
          """
          Creates a sparse module with the same defining data as self.
  
          N.B. This function is for internal use only! See sparse_module for
          use.
  
          EXAMPLES::
  
              sage: M = FreeModule(Integers(8),3)
              sage: S = FreeModule(Integers(8),3, sparse=True)
              sage: M._sparse_module() is S
              True
          """
          A = self.ambient_module().sparse_module()
          return A.span(self.basis())
  
      def _an_element_(self):
          """
          Returns an arbitrary element of a free module.
  
          EXAMPLES::
  
              sage: V = VectorSpace(QQ,2)
              sage: V._an_element_()
              (1, 0)
              sage: U = V.submodule([[1,0]])
              sage: U._an_element_()
              (1, 0)
              sage: W = V.submodule([])
              sage: W._an_element_()
              (0, 0)
          """
          try:
              return self.gen(0)
          except ValueError:
              return self(0)
  
      def some_elements(self):
          r"""
          Return some elements of this free module.
  
          See :class:`TestSuite` for a typical use case.
  
          OUTPUT:
  
          An iterator.
  
          EXAMPLES::
  
              sage: F = FreeModule(ZZ, 2)
              sage: tuple(F.some_elements())
              ((1, 0),
               (1, 1),
               (0, 1),
               (-1, 2),
               (-2, 3),
               ...
               (-49, 50))
  
              sage: F = FreeModule(QQ, 3)
              sage: tuple(F.some_elements())
              ((1, 0, 0),
               (1/2, 1/2, 1/2),
               (1/2, -1/2, 2),
               (-2, 0, 1),
               (-1, 42, 2/3),
               (-2/3, 3/2, -3/2),
               (4/5, -4/5, 5/4),
               ...
               (46/103823, -46/103823, 103823/46))
  
              sage: F = FreeModule(SR, 2)
              sage: tuple(F.some_elements())
              ((1, 0), (some_variable, some_variable))
          """
          yield self.an_element()
          yield self.base().an_element() * sum(self.gens())
          some_elements_base = iter(self.base().some_elements())
          n = self.degree()
          while True:
              L = list(islice(some_elements_base, int(n)))
              if len(L) != n:
                  return
              try:
                  yield self(L)
              except (TypeError, ValueError):
                  pass
  
      def _element_constructor_(self, x, coerce=True, copy=True, check=True):
          r"""
          Create an element of this free module from ``x``.
  
          The ``coerce`` and ``copy`` arguments are
          passed on to the underlying element constructor. If
          ``check`` is ``True``, confirm that the
          element specified by x does in fact lie in self.
  
          .. note::
  
             In the case of an inexact base ring (i.e. RDF), we don't
             verify that the element is in the subspace, even when
             ``check=True``, to account for numerical instability
             issues.
  
          EXAMPLES::
  
              sage: M = ZZ^4
              sage: M([1,-1,0,1])  #indirect doctest
              (1, -1, 0, 1)
              sage: M(0)
              (0, 0, 0, 0)
  
          ::
  
              sage: N = M.submodule([[1,0,0,0], [0,1,1,0]])
              sage: N([1,1,1,0])
              (1, 1, 1, 0)
              sage: N((3,-2,-2,0))
              (3, -2, -2, 0)
              sage: N((0,0,0,1))
              Traceback (most recent call last):
              ...
              TypeError: element (0, 0, 0, 1) is not in free module
  
          Beware that using check=False can create invalid results::
  
              sage: N((0,0,0,1), check=False)
              (0, 0, 0, 1)
              sage: N((0,0,0,1), check=False) in N
              True
          """
          if (isinstance(x, (int, sage.rings.integer.Integer)) and
              x == 0):
              return self.zero_vector()
          elif isinstance(x, free_module_element.FreeModuleElement):
              if x.parent() is self:
                  if copy:
                      return x.__copy__()
                  else:
                      return x
              x = x.list()
          if check and self.coordinate_ring().is_exact():
              if isinstance(self, FreeModule_ambient):
                  return self.element_class(self, x, coerce, copy)
              try:
                  c = self.coordinates(x)
                  R = self.base_ring()
                  for d in c:
                      if d not in R:
                          raise ArithmeticError
              except ArithmeticError:
                  raise TypeError("element {!r} is not in free module".format(x))
          return self.element_class(self, x, coerce, copy)
  
      def __richcmp__(self, other, op):
          r"""
          Rich comparison via containment in the same ambient space.
  
          Two modules compare if their ambient module/space is equal.
          Ambient spaces are equal if they have the same
          base ring, rank and inner product matrix.
  
          EXAMPLES:
  
          We compare rank three free modules over the integers,
          rationals, and complex numbers. Note the free modules
          ``QQ^3`` (and hence ``ZZ^3``) and ``CC^3`` are incomparable
          because of the different ambient vector spaces::
  
              sage: QQ^3 <= CC^3
              False
              sage: CC^3 <= QQ^3
              False
              sage: QQ^3 <= QQ^3
              True
  
              sage: QQ^3 <= ZZ^3
              False
              sage: ZZ^3 <= QQ^3
              True
              sage: ZZ^3 <= CC^3
              False
              sage: CC^3 <= ZZ^3
              False
  
          Comparison with a submodule::
  
              sage: A = QQ^3
              sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
              sage: V
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              sage: V <= A
              True
              sage: A <= V
              False
              sage: L1 = span([[1,2,3], [5,6,7], [8,9,10]], ZZ)
              sage: L2 = span([[2,4,6], [10,12,14], [16,18,20]], ZZ)
              sage: 2*L1 <= L2
              True
              sage: L2 <= L1
              True
              sage: L1 <= L2
              False
  
          More exotic comparisons::
  
              sage: R1 = ZZ[sqrt(2)]
              sage: F1 = R1^3
              sage: V1 = F1.span([[sqrt(2),sqrt(2),0]])
              sage: F2 = ZZ^3
              sage: V2 = F2.span([[2,2,0]])
              sage: V2 <= V1  # Different ambient vector spaces
              False
              sage: V1 <= V2
              False
              sage: R2 = GF(5)[x]
              sage: F3 = R2^3
              sage: V3 = F3.span([[x^5-1,1+x+x^2+x^3+x^4,0]])
              sage: W3 = F3.span([[1,1,0],[0,4,0]])
              sage: V3 <= W3
              True
              sage: W3 <= V3
              False
  
          We compare a one dimensional space to a two dimensional space::
  
              sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
              sage: M = span([[5,6,7]], QQ)
              sage: M <= V
              True
              sage: V <= M
              False
  
          We test that :trac:`5525` is fixed::
  
              sage: A = (QQ^1).span([[1/3]],ZZ); B = (QQ^1).span([[1]],ZZ)
              sage: A.intersection(B)
              Free module of degree 1 and rank 1 over Integer Ring
              Echelon basis matrix:
              [1]
  
          We create the module `\ZZ^3`, and the submodule generated by
          one vector `(1,1,0)`, and check whether certain elements are
          in the submodule::
  
              sage: R = FreeModule(ZZ, 3)
              sage: V = R.submodule([R.gen(0) + R.gen(1)])
              sage: R.gen(0) + R.gen(1) in V
              True
              sage: R.gen(0) + 2*R.gen(1) in V
              False
  
              sage: w = (1/2)*(R.gen(0) + R.gen(1))
              sage: w
              (1/2, 1/2, 0)
              sage: w.parent()
              Vector space of dimension 3 over Rational Field
              sage: w in V
              False
              sage: V.coordinates(w)
              [1/2]
  
          TESTS::
  
              sage: QQ^3 < ZZ^3
              False
              sage: ZZ^3 < QQ^3
              True
              sage: ZZ^3 < CC^3
              False
              sage: CC^3 < ZZ^3
              False
  
          Comparison with a submodule::
  
              sage: A = QQ^3
              sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
              sage: V
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              sage: V < A
              True
              sage: A < V
              False
              sage: L1 = span([[1,2,3], [5,6,7], [8,9,10]], ZZ)
              sage: L2 = span([[2,4,6], [10,12,14], [16,18,20]], ZZ)
              sage: 2*L1 < L2
              False
              sage: L2 < L1
              True
              sage: L1 < L2
              False
  
          We compare a `\ZZ`-module to a one-dimensional space::
  
              sage: V = span([[5,6,7]], ZZ).scale(1/11);  V
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [5/11 6/11 7/11]
              sage: M = span([[5,6,7]], QQ)
              sage: V < M
              True
              sage: M < V
              False
  
          We compare rank three free modules over the rationals and
          complex numbers::
  
              sage: QQ^3 >= CC^3
              False
              sage: CC^3 >= QQ^3
              False
              sage: QQ^3 >= QQ^3
              True
  
          Comparison with a submodule::
  
              sage: A = QQ^3
              sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
              sage: V
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              sage: V >= A
              False
              sage: A >= V
              True
              sage: L1 = span([[1,2,3], [5,6,7], [8,9,10]], ZZ)
              sage: L2 = span([[2,4,6], [10,12,14], [16,18,20]], ZZ)
              sage: 2*L1 >= L2
              True
              sage: L2 >= L1
              False
              sage: L1 >= L2
              True
  
          We compare rank three free modules over the rationals and
          complex numbers::
  
              sage: QQ^3 > CC^3
              False
              sage: CC^3 > QQ^3
              False
              sage: QQ^3 > QQ^3
              False
  
          Comparison with a submodule::
  
              sage: A = QQ^3
              sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
              sage: V
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              sage: V > A
              False
              sage: A > V
              True
              sage: L1 = span([[1,2,3], [5,6,7], [8,9,10]], ZZ)
              sage: L2 = span([[2,4,6], [10,12,14], [16,18,20]], ZZ)
              sage: 2*L1 > L2
              False
              sage: L2 > L1
              False
              sage: L1 > L2
              True
          """
          if self is other:
              return rich_to_bool(op, 0)
          if not isinstance(other, FreeModule_generic):
              return NotImplemented
          # Check equality first if needed
          if op == op_EQ:
              return self._eq(other)
          if op == op_NE:
              return not self._eq(other)
          if op == op_LE:
              return self.is_submodule(other)
          if op == op_GE:
              return other.is_submodule(self)
          if op == op_LT:
              return (not self._eq(other)) and self.is_submodule(other)
          if op == op_GT:
              return (not self._eq(other)) and other.is_submodule(self)
  
      def _eq(self, other):
          r"""
          Return if ``self`` is equal to ``other``.
  
          Ambient spaces are considered equal if they have the same
          rank, basering and inner product matrix.
  
          Modules in the same ambient space are partially ordered by inclusion.
  
          EXAMPLES::
  
              sage: L = IntegralLattice("U")
              sage: L is IntegralLattice("U")
              False
              sage: L._eq(IntegralLattice("U"))
              True
          """
          if self.rank() != other.rank():
              return False
          if self.base_ring() != other.base_ring():
              return False
          # We do not want to create an inner product matrix in memory if
          # self and other use the dot product
          if not (self._inner_product_is_dot_product()
                  and other._inner_product_is_dot_product()):
              # This only affects free_quadratic_modules
              if self.inner_product_matrix() != other.inner_product_matrix():
                  return False
          from sage.modules.quotient_module import FreeModule_ambient_field_quotient
          lq = isinstance(self, FreeModule_ambient_field_quotient)
          rq = isinstance(other, FreeModule_ambient_field_quotient)
          if lq or rq:
              # if the relations agree we continue with the covers.
              if lq:
                  lx = self.relations()
                  self = self.cover()
              else:
                  lx = self.zero_submodule()
              if rq:
                  rx = other.relations()
                  other = other.cover()
              else:
                  rx = other.zero_submodule()
              if lx != rx:
                  return False
          if isinstance(self, FreeModule_ambient) and isinstance(other, FreeModule_ambient):
              return True
          # self and other are not ambient.
          # but they are contained in the same ambient space
  
          # We use self.echelonized_basis_matrix() == other.echelonized_basis_matrix()
          # with the matrix to avoid a circular reference.
          from sage.rings.integer_ring import IntegerRing
          if self.base_ring().is_field() or self.base_ring() is IntegerRing():
              # We know that the Hermite normal form is unique here.
              return self.echelonized_basis_matrix() == other.echelonized_basis_matrix()
          return self.is_submodule(other) and other.is_submodule(self)
  
      def is_submodule(self, other):
          r"""
          Return ``True`` if ``self`` is a submodule of ``other``.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ,3)
              sage: V = M.ambient_vector_space()
              sage: X = V.span([[1/2,1/2,0],[1/2,0,1/2]], ZZ)
              sage: Y = V.span([[1,1,1]], ZZ)
              sage: N = X + Y
              sage: M.is_submodule(X)
              False
              sage: M.is_submodule(Y)
              False
              sage: Y.is_submodule(M)
              True
              sage: N.is_submodule(M)
              False
              sage: M.is_submodule(N)
              True
  
              sage: M = FreeModule(ZZ,2)
              sage: M.is_submodule(M)
              True
              sage: N = M.scale(2)
              sage: N.is_submodule(M)
              True
              sage: M.is_submodule(N)
              False
              sage: N = M.scale(1/2)
              sage: N.is_submodule(M)
              False
              sage: M.is_submodule(N)
              True
  
          Since :meth:`basis` is not implemented in general, submodule
          testing does not work for all PID's. However, trivial cases are
          already used (and useful) for coercion, e.g.::
  
              sage: QQ(1/2) * vector(ZZ['x']['y'],[1,2,3,4])
              (1/2, 1, 3/2, 2)
              sage: vector(ZZ['x']['y'],[1,2,3,4]) * QQ(1/2)
              (1/2, 1, 3/2, 2)
  
          TESTS::
  
              M = QQ^3 / [[1,2,3]]
              V = QQ^2
              V.is_submodule(M)
              False
  
              sage: M1 = QQ^3 / [[1,2,3]]
              sage: V1 = span(QQ,[(1,0,0)]) + M1.relations()
              sage: M2 = V1 / M1.relations()
              sage: M2.is_submodule(M1)  # Different ambient vector spaces
              False
          """
          if self is other:
              return True
          if not isinstance(other, FreeModule_generic):
              return False
          # Removing this try-except block changes the behavior of
          #   is_submodule for (QQ^2).is_submodule(CC^2)
          try:
              if self.ambient_vector_space() != other.ambient_vector_space():
                  return False
              if other is other.ambient_vector_space():
                  return True
          except AttributeError:
              # Not all free modules have an ambient_vector_space.
              pass
          from sage.modules.quotient_module import FreeModule_ambient_field_quotient
          if isinstance(other, FreeModule_ambient_field_quotient):
              #if the relations agree we continue with the covers.
              if isinstance(self, FreeModule_ambient_field_quotient):
                  if other.relations() != self.relations():
                      return False
                  self = self.cover()
              else:
                  if other.relations() != 0:
                      return False
              other = other.cover()
  
          if other.rank() < self.rank():
              return False
          if other.degree() != self.degree():
              return False
          if self._inner_product_is_dot_product() and other._inner_product_is_dot_product():
              pass
          else:
              if self.inner_product_matrix() != other.inner_product_matrix():
                  return False
          R = self.base_ring()
          S = other.base_ring()
          if R != S:
              try:
                  if not R.is_subring(S):
                      return False
              except NotImplementedError:
                  if not R.fraction_field().is_subring(S):
                      raise NotImplementedError("could not determine if %s is a "
                                                "subring of %s" % (R, S))
          # now R is a subring of S
          if other.is_ambient() and S.is_field():
              return True
          try:
              M = other.basis_matrix().solve_left(self.basis_matrix())
          except ValueError:
              return False
          except TypeError:
              # only if solve_left does not eat a matrix
              # else this is far to inefficient
              try:
                  M = [list(other.basis_matrix().solve_left(self.basis_matrix()[i])) for i in range(self.basis_matrix().nrows())]
              except ValueError:
                  return False
              from sage.misc.flatten import flatten
              return all(x in S for x in flatten(M))
          return all(x in S for x in M.list())
  
      def __iter__(self):
          """
          Return iterator over the elements of this free module.
  
          EXAMPLES::
  
              sage: V = VectorSpace(GF(4,'a'),2)
              sage: [x for x in V]
              [(0, 0), (a, 0), (a + 1, 0), (1, 0), (0, a), (a, a), (a + 1, a), (1, a), (0, a + 1), (a, a + 1), (a + 1, a + 1), (1, a + 1), (0, 1), (a, 1), (a + 1, 1), (1, 1)]
  
          ::
  
              sage: W = V.subspace([V([1,1])])
              sage: [x for x in W]
              [(0, 0), (a, a), (a + 1, a + 1), (1, 1)]
  
          TESTS::
  
              sage: V = VectorSpace(GF(2,'a'),2)
              sage: V.list()
              [(0, 0), (1, 0), (0, 1), (1, 1)]
          """
          G = self.gens()
          if not G:
              yield self(0)
              return
          R     = self.base_ring()
          iters = [iter(R) for _ in range(len(G))]
          for x in iters:
              next(x)     # put at 0
          zero  = R(0)
          v = [zero for _ in range(len(G))]
          n = 0
          z = self(0)
          yield z
          while n < len(G):
              try:
                  v[n] = next(iters[n])
                  yield self.linear_combination_of_basis(v)
                  n = 0
              except StopIteration:
                  iters[n] = iter(R)  # reset
                  next(iters[n])     # put at 0
                  v[n] = zero
                  n += 1
  
      def cardinality(self):
          r"""
          Return the cardinality of the free module.
  
          OUTPUT:
  
          Either an integer or ``+Infinity``.
  
          EXAMPLES::
  
              sage: k.<a> = FiniteField(9)
              sage: V = VectorSpace(k,3)
              sage: V.cardinality()
              729
              sage: W = V.span([[1,2,1],[0,1,1]])
              sage: W.cardinality()
              81
              sage: R = IntegerModRing(12)
              sage: M = FreeModule(R,2)
              sage: M.cardinality()
              144
              sage: (QQ^3).cardinality()
              +Infinity
  
          TESTS:
  
          Check that :trac:`22987` is fixed::
  
              sage: VectorSpace(QQ, 0).cardinality()
              1
          """
          if not self.rank():
              return sage.rings.integer.Integer(1)
          return self.base_ring().cardinality() ** self.rank()
  
      __len__ = cardinality # for backward compatibility
  
      def ambient_module(self):
          """
          Return the ambient module associated to this module.
  
          EXAMPLES::
  
              sage: R.<x,y> = QQ[]
              sage: M = FreeModule(R,2)
              sage: M.ambient_module()
              Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y over Rational Field
  
          ::
  
              sage: V = FreeModule(QQ, 4).span([[1,2,3,4], [1,0,0,0]]); V
              Vector space of degree 4 and dimension 2 over Rational Field
              Basis matrix:
              [  1   0   0   0]
              [  0   1 3/2   2]
              sage: V.ambient_module()
              Vector space of dimension 4 over Rational Field
          """
          return FreeModule(self.base_ring(), self.degree())
  
      def basis(self):
          """
          Return the basis of this module.
  
          EXAMPLES::
  
              sage: FreeModule(Integers(12),3).basis()
              [
              (1, 0, 0),
              (0, 1, 0),
              (0, 0, 1)
              ]
          """
          raise NotImplementedError
  
      def gens(self):
          """
          Return a tuple of basis elements of ``self``.
  
          EXAMPLES::
  
              sage: FreeModule(Integers(12),3).gens()
              ((1, 0, 0), (0, 1, 0), (0, 0, 1))
          """
          return tuple(self.basis())
  
      def basis_matrix(self, ring=None):
          """
          Return the matrix whose rows are the basis for this free module.
  
          INPUT:
  
          - ``ring`` -- (default: ``self.coordinate_ring()``) a ring over
            which the matrix is defined
  
          EXAMPLES::
  
              sage: FreeModule(Integers(12),3).basis_matrix()
              [1 0 0]
              [0 1 0]
              [0 0 1]
  
          ::
  
              sage: M = FreeModule(GF(7),3).span([[2,3,4],[1,1,1]]); M
              Vector space of degree 3 and dimension 2 over Finite Field of size 7
              Basis matrix:
              [1 0 6]
              [0 1 2]
              sage: M.basis_matrix()
              [1 0 6]
              [0 1 2]
  
          ::
  
              sage: M = FreeModule(GF(7),3).span_of_basis([[2,3,4],[1,1,1]])
              sage: M.basis_matrix()
              [2 3 4]
              [1 1 1]
  
          ::
  
              sage: M = FreeModule(QQ,2).span_of_basis([[1,-1],[1,0]]); M
              Vector space of degree 2 and dimension 2 over Rational Field
              User basis matrix:
              [ 1 -1]
              [ 1  0]
              sage: M.basis_matrix()
              [ 1 -1]
              [ 1  0]
  
          TESTS:
  
          See :trac:`3699`::
  
              sage: K = FreeModule(ZZ, 2000)
              sage: I = K.basis_matrix()
  
          See :trac:`17585`::
  
              sage: ((ZZ^2)*2).basis_matrix().parent()
              Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
              sage: ((ZZ^2)*2).basis_matrix(RDF).parent()
              Full MatrixSpace of 2 by 2 dense matrices over Real Double Field
  
              sage: M = (ZZ^2)*(1/2)
              sage: M.basis_matrix()
              [1/2   0]
              [  0 1/2]
              sage: M.basis_matrix().parent()
              Full MatrixSpace of 2 by 2 dense matrices over Rational Field
              sage: M.basis_matrix(QQ).parent()
              Full MatrixSpace of 2 by 2 dense matrices over Rational Field
              sage: M.basis_matrix(ZZ)
              Traceback (most recent call last):
              ...
              TypeError: matrix has denominators so can...t change to ZZ
          """
          try:
              A = self.__basis_matrix
          except AttributeError:
              MAT = sage.matrix.matrix_space.MatrixSpace(self.coordinate_ring(),
                              len(self.basis()), self.degree(),
                              sparse = self.is_sparse())
              if self.is_ambient():
                  A = MAT.identity_matrix()
              else:
                  A = MAT(self.basis())
              A.set_immutable()
              self.__basis_matrix = A
          if ring is None or ring is A.base_ring():
              return A
          else:
              return A.change_ring(ring)
  
      def echelonized_basis_matrix(self):
          """
          The echelonized basis matrix (not implemented for this module).
  
          This example works because M is an ambient module. Submodule
          creation should exist for generic modules.
  
          EXAMPLES::
  
              sage: R = IntegerModRing(12)
              sage: S.<x,y> = R[]
              sage: M = FreeModule(S,3)
              sage: M.echelonized_basis_matrix()
              [1 0 0]
              [0 1 0]
              [0 0 1]
  
          TESTS::
  
              sage: from sage.modules.free_module import FreeModule_generic
              sage: FreeModule_generic.echelonized_basis_matrix(M)
              Traceback (most recent call last):
              ...
              NotImplementedError
          """
          raise NotImplementedError
  
      def matrix(self):
          """
          Return the basis matrix of this module, which is the matrix whose
          rows are a basis for this module.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 2)
              sage: M.matrix()
              [1 0]
              [0 1]
              sage: M.submodule([M.gen(0) + M.gen(1), M.gen(0) - 2*M.gen(1)]).matrix()
              [1 1]
              [0 3]
          """
          return self.basis_matrix()
  
      def direct_sum(self, other):
          """
          Return the direct sum of ``self`` and ``other`` as a free module.
  
          EXAMPLES::
  
              sage: V = (ZZ^3).span([[1/2,3,5], [0,1,-3]]); V
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [1/2   0  14]
              [  0   1  -3]
              sage: W = (ZZ^3).span([[1/2,4,2]]); W
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [1/2   4   2]
              sage: V.direct_sum(W)
              Free module of degree 6 and rank 3 over Integer Ring
              Echelon basis matrix:
              [1/2   0  14   0   0   0]
              [  0   1  -3   0   0   0]
              [  0   0   0 1/2   4   2]
          """
          if not is_FreeModule(other):
              raise TypeError("other must be a free module")
          if other.base_ring() != self.base_ring():
              raise TypeError("base rings of self and other must be the same")
          return self.basis_matrix().block_sum(other.basis_matrix()).row_module(self.base_ring())
  
      def coordinates(self, v, check=True):
          """
          Write `v` in terms of the basis for self.
  
          INPUT:
  
          - ``v`` -- vector
  
          - ``check`` -- bool (default: True); if True, also verify that
             `v` is really in self.
  
          OUTPUT: list
  
          Returns a list `c` such that if `B` is the basis
          for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 2); M0,M1=M.gens()
              sage: W = M.submodule([M0 + M1, M0 - 2*M1])
              sage: W.coordinates(2*M0-M1)
              [2, -1]
          """
          return self.coordinate_vector(v, check=check).list()
  
      def coordinate_vector(self, v, check=True):
          """
          Return the vector whose coefficients give `v` as a linear
          combination of the basis for self.
  
          INPUT:
  
          - ``v`` -- vector
  
          - ``check`` -- bool (default: True); if True, also verify that
             `v` is really in self.
  
          OUTPUT: list
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 2); M0,M1=M.gens()
              sage: W = M.submodule([M0 + M1, M0 - 2*M1])
              sage: W.coordinate_vector(2*M0 - M1)
              (2, -1)
          """
          raise NotImplementedError
  
      def coordinate_module(self, V):
          r"""
          Suppose ``V`` is a submodule of ``self`` (or a module commensurable
          with ``self``), and that ``self`` is a free module over `R` of rank
          `n`. Let `\phi` be the map from ``self`` to
          `R^n` that sends the basis vectors of ``self`` in order to the
          standard basis of `R^n`. This function returns the image
          `\phi(V)`.
  
          .. WARNING::
  
             If there is no integer `d` such that `dV` is a submodule
             of ``self``, then this function will give total nonsense.
  
          EXAMPLES:
  
          We illustrate this function with some
          `\ZZ`-submodules of `\QQ^3`::
  
              sage: V = (ZZ^3).span([[1/2,3,5], [0,1,-3]])
              sage: W = (ZZ^3).span([[1/2,4,2]])
              sage: V.coordinate_module(W)
              Free module of degree 2 and rank 1 over Integer Ring
              User basis matrix:
              [1 4]
              sage: V.0 + 4*V.1
              (1/2, 4, 2)
  
          In this example, the coordinate module isn't even in `\ZZ^3`::
  
              sage: W = (ZZ^3).span([[1/4,2,1]])
              sage: V.coordinate_module(W)
              Free module of degree 2 and rank 1 over Integer Ring
              User basis matrix:
              [1/2   2]
  
          The following more elaborate example illustrates using this
          function to write a submodule in terms of integral cuspidal modular
          symbols::
  
              sage: M = ModularSymbols(54)
              sage: S = M.cuspidal_subspace()
              sage: K = S.integral_structure(); K
              Free module of degree 19 and rank 8 over Integer Ring
              Echelon basis matrix:
              [ 0  1  0  0 -1  0  0  0  0  0  0  0  0  0  0  0  0  0  0]
              ...
              sage: L = M[0].integral_structure(); L
              Free module of degree 19 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 0  1  1  0 -2  1 -1  1 -1 -2  2  0  0  0  0  0  0  0  0]
              [ 0  0  3  0 -3  2 -1  2 -1 -4  2 -1 -2  1  2  0  0 -1  1]
              sage: K.coordinate_module(L)
              Free module of degree 8 and rank 2 over Integer Ring
              User basis matrix:
              [ 1  1  1 -1  1 -1  0  0]
              [ 0  3  2 -1  2 -1 -1 -2]
              sage: K.coordinate_module(L).basis_matrix() * K.basis_matrix()
              [ 0  1  1  0 -2  1 -1  1 -1 -2  2  0  0  0  0  0  0  0  0]
              [ 0  0  3  0 -3  2 -1  2 -1 -4  2 -1 -2  1  2  0  0 -1  1]
          """
          if not is_FreeModule(V):
              raise ValueError("V must be a free module")
          A = self.basis_matrix()
          A = A.matrix_from_columns(A.pivots()).transpose()
          B = V.basis_matrix()
          B = B.matrix_from_columns(self.basis_matrix().pivots()).transpose()
          S = A.solve_right(B).transpose()
          return (self.base_ring()**S.ncols()).span_of_basis(S.rows())
  
      def degree(self):
          """
          Return the degree of this free module. This is the dimension of the
          ambient vector space in which it is embedded.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 10)
              sage: W = M.submodule([M.gen(0), 2*M.gen(3) - M.gen(0), M.gen(0) + M.gen(3)])
              sage: W.degree()
              10
              sage: W.rank()
              2
          """
          return self.__degree
  
      def dimension(self):
          """
          Return the dimension of this free module.
  
          EXAMPLES::
  
              sage: M = FreeModule(FiniteField(19), 100)
              sage: W = M.submodule([M.gen(50)])
              sage: W.dimension()
              1
          """
          return self.rank()
  
      def codimension(self):
          """
          Return the codimension of this free module, which is the
          dimension of the ambient space minus the dimension of this
          free module.
  
          EXAMPLES::
  
              sage: M = Matrix(3, 4, range(12))
              sage: V = M.left_kernel(); V
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [ 1 -2  1]
              sage: V.dimension()
              1
              sage: V.codimension()
              2
  
          The codimension of an ambient space is always zero::
  
              sage: (QQ^10).codimension()
              0
          """
          return self.degree() - self.rank()
  
      def discriminant(self):
          """
          Return the discriminant of this free module.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 3)
              sage: M.discriminant()
              1
              sage: W = M.span([[1,2,3]])
              sage: W.discriminant()
              14
              sage: W2 = M.span([[1,2,3], [1,1,1]])
              sage: W2.discriminant()
              6
          """
          return self.gram_matrix().determinant()
  
      def base_field(self):
          """
          Return the base field, which is the fraction field of the base ring
          of this module.
  
          EXAMPLES::
  
              sage: FreeModule(GF(3), 2).base_field()
              Finite Field of size 3
              sage: FreeModule(ZZ, 2).base_field()
              Rational Field
              sage: FreeModule(PolynomialRing(GF(7),'x'), 2).base_field()
              Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 7
          """
          return self.base_ring().fraction_field()
  
      def coordinate_ring(self):
          """
          Return the ring over which the entries of the vectors are
          defined.
  
          This is the same as :meth:`base_ring` unless an explicit basis
          was given over the fraction field.
  
          EXAMPLES::
  
              sage: M = ZZ^2
              sage: M.coordinate_ring()
              Integer Ring
  
          ::
  
              sage: M = (ZZ^2) * (1/2)
              sage: M.base_ring()
              Integer Ring
              sage: M.coordinate_ring()
              Rational Field
  
          ::
  
              sage: R.<x> = QQ[]
              sage: L = R^2
              sage: L.coordinate_ring()
              Univariate Polynomial Ring in x over Rational Field
              sage: L.span([(x,0), (1,x)]).coordinate_ring()
              Univariate Polynomial Ring in x over Rational Field
              sage: L.span([(x,0), (1,1/x)]).coordinate_ring()
              Fraction Field of Univariate Polynomial Ring in x over Rational Field
              sage: L.span([]).coordinate_ring()
              Univariate Polynomial Ring in x over Rational Field
          """
          return self.__coordinate_ring
  
      def free_module(self):
          """
          Return this free module. (This is used by the
          ``FreeModule`` functor, and simply returns self.)
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 3)
              sage: M.free_module()
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
          """
          return self
  
      def gen(self, i=0):
          """
          Return the `i`-th generator for ``self``.
  
          Here `i` is between 0 and rank - 1, inclusive.
  
          INPUT:
  
          - `i` -- an integer (default 0)
  
          OUTPUT: `i`-th basis vector for ``self``.
  
          EXAMPLES::
  
              sage: n = 5
              sage: V = QQ^n
              sage: B = [V.gen(i) for i in range(n)]
              sage: B
              [(1, 0, 0, 0, 0),
              (0, 1, 0, 0, 0),
              (0, 0, 1, 0, 0),
              (0, 0, 0, 1, 0),
              (0, 0, 0, 0, 1)]
              sage: V.gens() == tuple(B)
              True
  
          TESTS::
  
              sage: (QQ^3).gen(4/3)
              Traceback (most recent call last):
              ...
              TypeError: unable to convert rational 4/3 to an integer
          """
          if i < 0 or i >= self.rank():
              raise ValueError("Generator %s not defined." % i)
          return self.basis()[i]
  
      def gram_matrix(self):
          r"""
          Return the gram matrix associated to this free module, defined to
          be `G = B*A*B.transpose()`, where A is the inner product matrix
          (induced from the ambient space), and B the basis matrix.
  
          EXAMPLES::
  
              sage: V = VectorSpace(QQ,4)
              sage: u = V([1/2,1/2,1/2,1/2])
              sage: v = V([0,1,1,0])
              sage: w = V([0,0,1,1])
              sage: M = span([u,v,w], ZZ)
              sage: M.inner_product_matrix() == V.inner_product_matrix()
              True
              sage: L = M.submodule_with_basis([u,v,w])
              sage: L.inner_product_matrix() == M.inner_product_matrix()
              True
              sage: L.gram_matrix()
              [1 1 1]
              [1 2 1]
              [1 1 2]
          """
          if self.is_ambient():
              return sage.matrix.matrix_space.MatrixSpace(self.base_ring(), self.degree(), sparse=True)(1)
          else:
              if self._gram_matrix is None:
                  B = self.basis_matrix()
                  self._gram_matrix = B*B.transpose()
              return self._gram_matrix
  
      def has_user_basis(self):
          """
          Return ``True`` if the basis of this free module is
          specified by the user, as opposed to being the default echelon
          form.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.subspace([[2,'1/2', 1]])
              sage: W.has_user_basis()
              False
              sage: W = V.subspace_with_basis([[2,'1/2',1]])
              sage: W.has_user_basis()
              True
          """
          return False
  
      def hom(self, im_gens, codomain=None, **kwds):
          """
          Override the hom method to handle the case of morphisms given by left-multiplication
          of a matrix and the codomain is not given.
  
          EXAMPLES::
  
              sage: W = ZZ^2; W.hom(matrix(1, [1, 2]), side="right")
              Free module morphism defined as left-multiplication by the matrix
              [1 2]
              Domain: Ambient free module of rank 2 over the principal ideal domain Integer Ring
              Codomain: Ambient free module of rank 1 over the principal ideal domain Integer Ring
              sage: V = QQ^2; V.hom(identity_matrix(2), side="right")
              Vector space morphism represented as left-multiplication by the matrix:
              [1 0]
              [0 1]
              Domain: Vector space of dimension 2 over Rational Field
              Codomain: Vector space of dimension 2 over Rational Field
          """
          from sage.structure.element import is_Matrix
          if codomain is None and is_Matrix(im_gens):
              side = kwds.get("side", "left")
              n = im_gens.nrows() if side == "right" else im_gens.ncols()
              from sage.categories.pushout import pushout
              R = pushout(self.base_ring(), im_gens.base_ring())
              codomain = R**n
          return super().hom(im_gens, codomain, **kwds)
  
      def inner_product_matrix(self):
          """
          Return the default identity inner product matrix associated to this
          module.
  
          By definition this is the inner product matrix of the ambient
          space, hence may be of degree greater than the rank of the module.
  
          TODO: Differentiate the image ring of the inner product from the
          base ring of the module and/or ambient space. E.g. On an integral
          module over ZZ the inner product pairing could naturally take
          values in ZZ, QQ, RR, or CC.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 3)
              sage: M.inner_product_matrix()
              [1 0 0]
              [0 1 0]
              [0 0 1]
          """
          return sage.matrix.matrix_space.MatrixSpace(self.base_ring(), self.degree(), sparse=True)(1)
  
      def _inner_product_is_dot_product(self):
          """
          Return whether or not the inner product on this module is induced
          by the dot product on the ambient vector space. This is used
          internally by the inner_product function for optimization.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 3)._inner_product_is_dot_product()
              True
              sage: FreeModule(ZZ, 3, inner_product_matrix=1)._inner_product_is_dot_product()
              True
              sage: FreeModule(ZZ, 2, inner_product_matrix=[1,0,-1,0])._inner_product_is_dot_product()
              False
  
          ::
  
              sage: M = FreeModule(QQ, 3)
              sage: M2 = M.span([[1,2,3]])
              sage: M2._inner_product_is_dot_product()
              True
          """
          return True
  
      def is_ambient(self):
          """
          Returns False since this is not an ambient free module.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 3).span([[1,2,3]]); M
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [1 2 3]
              sage: M.is_ambient()
              False
              sage: M = (ZZ^2).span([[1,0], [0,1]])
              sage: M
              Free module of degree 2 and rank 2 over Integer Ring
              Echelon basis matrix:
              [1 0]
              [0 1]
              sage: M.is_ambient()
              False
              sage: M == M.ambient_module()
              True
          """
          return False
  
      def is_dense(self):
          """
          Return ``True`` if the underlying representation of
          this module uses dense vectors, and False otherwise.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 2).is_dense()
              True
              sage: FreeModule(ZZ, 2, sparse=True).is_dense()
              False
          """
          return not self.is_sparse()
  
      def is_full(self):
          """
          Return ``True`` if the rank of this module equals its
          degree.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 2).is_full()
              True
              sage: M = FreeModule(ZZ, 2).span([[1,2]])
              sage: M.is_full()
              False
          """
          return self.rank() == self.degree()
  
      def is_finite(self):
          """
          Returns True if the underlying set of this free module is finite.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 2).is_finite()
              False
              sage: FreeModule(Integers(8), 2).is_finite()
              True
              sage: FreeModule(ZZ, 0).is_finite()
              True
          """
          return self.base_ring().is_finite() or self.rank() == 0
  
      def is_sparse(self):
          """
          Return ``True`` if the underlying representation of
          this module uses sparse vectors, and False otherwise.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 2).is_sparse()
              False
              sage: FreeModule(ZZ, 2, sparse=True).is_sparse()
              True
          """
          return self.__is_sparse
  
      def ngens(self):
          """
          Returns the number of basis elements of this free module.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 2).ngens()
              2
              sage: FreeModule(ZZ, 0).ngens()
              0
              sage: FreeModule(ZZ, 2).span([[1,1]]).ngens()
              1
          """
          try:
              return self.__ngens
          except AttributeError:
              self.__ngens = self.rank()
          return self.__ngens
  
      def nonembedded_free_module(self):
          """
          Returns an ambient free module that is isomorphic to this free
          module.
  
          Thus if this free module is of rank `n` over a ring
          `R`, then this function returns `R^n`, as an
          ambient free module.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 2).span([[1,1]]).nonembedded_free_module()
              Ambient free module of rank 1 over the principal ideal domain Integer Ring
          """
          return FreeModule(self.base_ring(), self.rank())
  
      def random_element(self, prob=1.0, *args, **kwds):
          """
          Returns a random element of self.
  
          INPUT:
  
          -- ``prob`` - float. Each coefficient will be set to zero with
             probability `1-prob`. Otherwise coefficients will be chosen
             randomly from base ring (and may be zero).
  
          -- ``*args, **kwds`` - passed on to ``random_element()`` function
             of base ring.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 2).span([[1, 1]])
              sage: v = M.random_element()
              sage: v.parent() is M
              True
              sage: v in M
              True
  
          Small entries are likely::
  
              sage: for i in [-2, -1, 0, 1, 2]:
              ....:     while vector([i, i]) != M.random_element():
              ....:         pass
  
          Large entries appear as well::
  
              sage: while abs(M.random_element()[0]) < 100:
              ....:     pass
  
          Passes extra positional or keyword arguments through::
  
              sage: all(i in range(5, 10) for i in M.random_element(1.0, 5, 10))
              True
          """
          rand = current_randstate().python_random().random
          R = self.base_ring()
          prob = float(prob)
          c = [0 if rand() > prob else R.random_element(*args, **kwds) for _ in range(self.rank())]
          return self.linear_combination_of_basis(c)
  
      def rank(self):
          """
          Return the rank of this free module.
  
          EXAMPLES::
  
              sage: FreeModule(Integers(6), 10000000).rank()
              10000000
              sage: FreeModule(ZZ, 2).span([[1,1], [2,2], [3,4]]).rank()
              2
          """
          return self.__rank
  
      def __bool__(self):
          """
          Return ``True`` if and only if the rank of this module is
          non-zero. In other words, this returns ``False`` for the zero
          module and ``True`` otherwise (apart from the exceptional case
          where the base ring is the zero ring).
  
          EXAMPLES::
  
              sage: bool(QQ^0)
              False
              sage: bool(QQ^1)
              True
              sage: M = Matrix(2, 3, range(6))
              sage: bool(M.right_kernel())
              True
              sage: bool(M.left_kernel())
              False
  
          When the base ring is the zero ring, we still look at the
          "rank" (which may not be mathematically meaningful)::
  
              sage: M = Integers(1)^4; M
              Ambient free module of rank 4 over Ring of integers modulo 1
              sage: M.rank()
              4
              sage: bool(M)
              True
              sage: M.cardinality()
              1
          """
          return bool(self.rank())
  
      __nonzero__ = __bool__
  
      def uses_ambient_inner_product(self):
          r"""
          Return ``True`` if the inner product on this module is
          the one induced by the ambient inner product.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 2)
              sage: W = M.submodule([[1,2]])
              sage: W.uses_ambient_inner_product()
              True
              sage: W.inner_product_matrix()
              [1 0]
              [0 1]
  
          ::
  
              sage: W.gram_matrix()
              [5]
          """
          return self.__uses_ambient_inner_product
  
      def zero_vector(self):
          """
          Returns the zero vector in this free module.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 2)
              sage: M.zero_vector()
              (0, 0)
              sage: M(0)
              (0, 0)
              sage: M.span([[1,1]]).zero_vector()
              (0, 0)
              sage: M.zero_submodule().zero_vector()
              (0, 0)
          """
          # Do *not* cache this -- it must be computed fresh each time, since
          # it is used by __call__ to make a new copy of the 0 element.
  
          return self.element_class(self, 0)
  
      @cached_method
      def zero(self):
          """
          Returns the zero vector in this free module.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 2)
              sage: M.zero()
              (0, 0)
              sage: M.span([[1,1]]).zero()
              (0, 0)
              sage: M.zero_submodule().zero()
              (0, 0)
              sage: M.zero_submodule().zero().is_mutable()
              False
          """
          res = self.element_class(self, 0)
          res.set_immutable()
          return res
  
      def are_linearly_dependent(self, vecs):
          """
          Return ``True`` if the vectors ``vecs`` are linearly dependent and
          ``False`` otherwise.
  
          EXAMPLES::
  
              sage: M = QQ^3
              sage: vecs = [M([1,2,3]), M([4,5,6])]
              sage: M.are_linearly_dependent(vecs)
              False
              sage: vecs.append(M([3,3,3]))
              sage: M.are_linearly_dependent(vecs)
              True
  
              sage: R.<x> = QQ[]
              sage: M = FreeModule(R, 2)
              sage: vecs = [M([x^2+1, x+1]), M([x+2, 2*x+1])]
              sage: M.are_linearly_dependent(vecs)
              False
              sage: vecs.append(M([-2*x+1, -2*x^2+1]))
              sage: M.are_linearly_dependent(vecs)
              True
          """
          from sage.matrix.constructor import matrix
          A = matrix(vecs)
          A.echelonize()
          return any(row.is_zero() for row in A.rows())
  
      def _magma_init_(self, magma):
          """
          EXAMPLES::
  
              sage: magma(QQ^9)                                   # optional - magma
              Full Vector space of degree 9 over Rational Field
              sage: (QQ^9)._magma_init_(magma)                    # optional - magma
              'RSpace(_sage_[...],9)'
  
          ::
  
              sage: magma(Integers(8)^2)                          # optional - magma
              Full RSpace of degree 2 over IntegerRing(8)
              sage: magma(FreeModule(QQ['x'], 2))                 # optional - magma
              Full RSpace of degree 2 over Univariate Polynomial Ring in x over Rational Field
  
          ::
  
              sage: A = matrix([[1,0],[0,-1]])
              sage: M = FreeModule(ZZ,2,inner_product_matrix=A); M
              Ambient free quadratic module of rank 2 over the principal ideal domain Integer Ring
              Inner product matrix:
              [ 1  0]
              [ 0 -1]
              sage: M._magma_init_(magma)                         # optional - magma
              'RSpace(_sage_[...],2,_sage_ref...)'
              sage: m = magma(M); m                               # optional - magma
              Full RSpace of degree 2 over Integer Ring
              Inner Product Matrix:
              [ 1  0]
              [ 0 -1]
              sage: m.Type()                                      # optional - magma
              ModTupRng
              sage: m.sage()                                      # optional - magma
              Ambient free quadratic module of rank 2 over the principal ideal domain Integer Ring
              Inner product matrix:
              [ 1  0]
              [ 0 -1]
              sage: m.sage() is M                               # optional - magma
              True
  
          Now over a field::
  
              sage: N = FreeModule(QQ,2,inner_product_matrix=A); N
              Ambient quadratic space of dimension 2 over Rational Field
              Inner product matrix:
              [ 1  0]
              [ 0 -1]
              sage: n = magma(N); n                                      # optional - magma
              Full Vector space of degree 2 over Rational Field
              Inner Product Matrix:
              [ 1  0]
              [ 0 -1]
              sage: n.Type()                                             # optional - magma
              ModTupFld
              sage: n.sage()                                           # optional - magma
              Ambient quadratic space of dimension 2 over Rational Field
              Inner product matrix:
              [ 1  0]
              [ 0 -1]
              sage: n.sage() is N                                      # optional - magma
              True
  
          How about some inexact fields::
  
              sage: v = vector(RR, [1, pi, 5/6])
              sage: F = v.parent()
              sage: M = magma(F); M # optional - magma
              Full Vector space of degree 3 over Real field of precision 15
              sage: M.Type() # optional - magma
              ModTupFld
              sage: m = M.sage(); m # optional - magma
              Vector space of dimension 3 over Real Field with 53 bits of precision
              sage: m is F # optional - magma
              True
  
          For interval fields, we can convert to Magma but there is no
          interval field in Magma so we cannot convert back::
  
              sage: v = vector(RealIntervalField(100), [1, pi, 0.125])
              sage: F = v.parent()
              sage: M = magma(v.parent()); M # optional - magma
              Full Vector space of degree 3 over Real field of precision 30
              sage: M.Type() # optional - magma
              ModTupFld
              sage: m = M.sage(); m # optional - magma
              Vector space of dimension 3 over Real Field with 100 bits of precision
              sage: m is F # optional - magma
              False
          """
          K = magma(self.base_ring())
          if not self._inner_product_is_dot_product():
              M = magma(self.inner_product_matrix())
              return "RSpace(%s,%s,%s)"%(K.name(), self.rank(), M._ref())
          else:
              return "RSpace(%s,%s)"%(K.name(), self.rank())
  
      def _macaulay2_(self, macaulay2=None):
          r"""
          EXAMPLES::
  
              sage: R = QQ^2
              sage: macaulay2(R) # optional - macaulay2
                2
              QQ
          """
          if macaulay2 is None:
              from sage.interfaces.macaulay2 import macaulay2
          if hasattr(self, '_inner_product_matrix'):
              raise NotImplementedError
          else:
              return macaulay2(self.base_ring())**self.rank()
  
  class FreeModule_generic_pid(FreeModule_generic):
      """
      Base class for all free modules over a PID.
      """
      def __init__(self, base_ring, rank, degree, sparse=False, coordinate_ring=None):
          """
          Create a free module over a PID.
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 2)
              Ambient free module of rank 2 over the principal ideal domain Integer Ring
              sage: FreeModule(PolynomialRing(GF(7),'x'), 2)
              Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Finite Field of size 7
          """
          # The first check should go away once everything is categorized...
          if base_ring not in PrincipalIdealDomains():
              raise TypeError("The base_ring must be a principal ideal domain.")
          super(FreeModule_generic_pid, self).__init__(base_ring, rank, degree,
                                                       sparse, coordinate_ring)
  
      def scale(self, other):
          """
          Return the product of this module by the number other, which is the
          module spanned by other times each basis vector.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 3)
              sage: M.scale(2)
              Free module of degree 3 and rank 3 over Integer Ring
              Echelon basis matrix:
              [2 0 0]
              [0 2 0]
              [0 0 2]
  
          ::
  
              sage: a = QQ('1/3')
              sage: M.scale(a)
              Free module of degree 3 and rank 3 over Integer Ring
              Echelon basis matrix:
              [1/3   0   0]
              [  0 1/3   0]
              [  0   0 1/3]
          """
          if other == 0:
              return self.zero_submodule()
          if other == 1 or other == -1:
              return self
          return self.span([v*other for v in self.basis()])
  
      def __radd__(self, other):
          """
          EXAMPLES::
  
              sage: int(0) + QQ^3
              Vector space of dimension 3 over Rational Field
              sage: sum([QQ^3, QQ^3])
              Vector space of degree 3 and dimension 3 over Rational Field
              Basis matrix:
              [1 0 0]
              [0 1 0]
              [0 0 1]
          """
          if other == 0:
              return self
          else:
              raise TypeError
  
      def __add__(self, other):
          r"""
          Return the sum of ``self`` and other, where both ``self`` and ``other`` must be
          submodules of the ambient vector space.
  
          EXAMPLES:
  
          We add two vector spaces::
  
              sage: V  = VectorSpace(QQ, 3)
              sage: W  = V.subspace([V([1,1,0])])
              sage: W2 = V.subspace([V([1,-1,0])])
              sage: W + W2
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [1 0 0]
              [0 1 0]
  
          We add two free `\ZZ`-modules.
  
          ::
  
              sage: M = FreeModule(ZZ, 3)
              sage: W = M.submodule([M([1,0,2])])
              sage: W2 = M.submodule([M([2,0,-4])])
              sage: W + W2
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [1 0 2]
              [0 0 8]
  
          We can also add free `\ZZ`-modules embedded
          non-integrally into an ambient space.
  
          ::
  
              sage: V = VectorSpace(QQ, 3)
              sage: W = M.span([1/2*V.0 - 1/3*V.1])
  
          Here the command ``M.span(...)`` creates the span of
          the indicated vectors over the base ring of `M`.
  
          ::
  
              sage: W2 = M.span([1/3*V.0 + V.1])
              sage: W + W2
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 1/6  7/3    0]
              [   0 11/3    0]
  
          We add two modules over `\ZZ`::
  
              sage: A = Matrix(ZZ, 3, 3, [3, 0, -1, 0, -2, 0, 0, 0, -2])
              sage: V = (A+2).kernel()
              sage: W = (A-3).kernel()
              sage: V+W
              Free module of degree 3 and rank 3 over Integer Ring
              Echelon basis matrix:
              [5 0 0]
              [0 1 0]
              [0 0 1]
  
          We add a module to 0::
  
              sage: ZZ^3 + 0
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
          """
          if not isinstance(other, FreeModule_generic):
              if other == 0:
                  return self
              raise TypeError("other (=%s) must be a free module"%other)
          if not (self.ambient_vector_space() == other.ambient_vector_space()):
              raise TypeError("ambient vector spaces must be equal")
          return self.span(self.basis() + other.basis())
  
      def _mul_(self, other, switch_sides=False):
          r"""
          Multiplication of the basis by ``other``.
  
          EXAMPLES::
  
              sage: A = ZZ^3
              sage: A * 3
              Free module of degree 3 and rank 3 over Integer Ring
              Echelon basis matrix:
              [3 0 0]
              [0 3 0]
              [0 0 3]
  
              sage: V = A.span([A([1,2,2]), A([-1,0,2])])
              sage: 2 * V
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 2  0 -4]
              [ 0  4  8]
  
              sage: m = matrix(3, range(9))
              sage: A * m
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 3  0 -3]
              [ 0  1  2]
              sage: m * A
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 3  0 -3]
              [ 0  1  2]
  
          TESTS:
  
          Check that :trac:`17705` is fixed::
  
              sage: V = GF(2)^2
              sage: W = V.subspace([[1, 0]])
              sage: x = matrix(GF(2), [[1, 1], [0, 1]])
              sage: W*x
              Vector space of degree 2 and dimension 1 over Finite Field of size 2
              Basis matrix:
              [1 1]
  
          """
          B = self.basis_matrix()
          B = other * B if switch_sides else B * other
          return self.span(B.rows())
  
      def index_in(self, other):
          """
          Return the lattice index [other:self] of ``self`` in other, as an
          element of the base field. When ``self`` is contained in other, the
          lattice index is the usual index. If the index is infinite, then
          this function returns infinity.
  
          EXAMPLES::
  
              sage: L1 = span([[1,2]], ZZ)
              sage: L2 = span([[3,6]], ZZ)
              sage: L2.index_in(L1)
              3
  
          Note that the free modules being compared need not be integral.
  
          ::
  
              sage: L1 = span([['1/2','1/3'], [4,5]], ZZ)
              sage: L2 = span([[1,2], [3,4]], ZZ)
              sage: L2.index_in(L1)
              12/7
              sage: L1.index_in(L2)
              7/12
              sage: L1.discriminant() / L2.discriminant()
              49/144
  
          The index of a lattice of infinite index is infinite.
  
          ::
  
              sage: L1 = FreeModule(ZZ, 2)
              sage: L2 = span([[1,2]], ZZ)
              sage: L2.index_in(L1)
              +Infinity
          """
          if not isinstance(other, FreeModule_generic):
              raise TypeError("other must be a free module")
  
          if self.ambient_vector_space() != other.ambient_vector_space():
              raise ArithmeticError("self and other must be embedded in the same ambient space.")
  
          if self.base_ring() != other.base_ring():
              raise NotImplementedError("lattice index only defined for modules over the same base ring.")
  
          if other.base_ring().is_field():
              if self == other:
                  return sage.rings.integer.Integer(1)
              else:
                  if self.is_subspace(other):
                      return sage.rings.infinity.infinity
              raise ArithmeticError("self must be contained in the vector space spanned by other.")
  
          C = [other.coordinates(b) for b in self.basis()]
  
          if self.rank() < other.rank():
              return sage.rings.infinity.infinity
  
          a = sage.matrix.matrix_space.MatrixSpace(self.base_field(), self.rank())(C).determinant()
          if sage.rings.integer_ring.is_IntegerRing(self.base_ring()):
              return a.abs()
          elif isinstance(self.base_ring, sage.rings.number_field.order.Order):
              return self.base_ring().ideal(a).norm()
          else:
              raise NotImplementedError
  
      def intersection(self, other):
          r"""
          Return the intersection of ``self`` and ``other``.
  
          EXAMPLES:
  
          We intersect two submodules one of which is clearly
          contained in the other::
  
              sage: A = ZZ^2
              sage: M1 = A.span([[1,1]])
              sage: M2 = A.span([[3,3]])
              sage: M1.intersection(M2)
              Free module of degree 2 and rank 1 over Integer Ring
              Echelon basis matrix:
              [3 3]
              sage: M1.intersection(M2) is M2
              True
  
          We intersection two submodules of `\ZZ^3` of rank
          `2`, whose intersection has rank `1`::
  
              sage: A = ZZ^3
              sage: M1 = A.span([[1,1,1], [1,2,3]])
              sage: M2 = A.span([[2,2,2], [1,0,0]])
              sage: M1.intersection(M2)
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [2 2 2]
  
          We compute an intersection of two `\ZZ`-modules that
          are not submodules of `\ZZ^2`::
  
              sage: A = ZZ^2
              sage: M1 = A.span([[1,2]]).scale(1/6)
              sage: M2 = A.span([[1,2]]).scale(1/15)
              sage: M1.intersection(M2)
              Free module of degree 2 and rank 1 over Integer Ring
              Echelon basis matrix:
              [1/3 2/3]
  
          We intersect a `\ZZ`-module with a `\QQ`-vector space::
  
              sage: A = ZZ^3
              sage: L = ZZ^3
              sage: V = QQ^3
              sage: W = L.span([[1/2,0,1/2]])
              sage: K = V.span([[1,0,1], [0,0,1]])
              sage: W.intersection(K)
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [1/2   0 1/2]
              sage: K.intersection(W)
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [1/2   0 1/2]
  
          We intersect two modules over the ring of integers of a number field::
  
              sage: L.<w> = NumberField(x^2 - x + 2)
              sage: OL = L.ring_of_integers()
              sage: V = L**3
              sage: W1 = V.span([[0,w/5,0], [1,0,-1/17]], OL)
              sage: W2 = V.span([[0,(1-w)/5,0]], OL)
              sage: W1.intersection(W2)
              Free module of degree 3 and rank 1 over Maximal Order in
               Number Field in w with defining polynomial x^2 - x + 2
              Echelon basis matrix:
              [  0 2/5   0]
  
          TESTS:
  
          Check that :trac:`24702` is fixed::
  
              sage: L = FreeQuadraticModule(ZZ,2,matrix.identity(2))
              sage: S1 = L.submodule([(1,0)])
              sage: S2 = L.submodule([(0,1)])
              sage: S1.intersection(S2).ambient_module() == S1.ambient_module()
              True
          """
          if not isinstance(other, FreeModule_generic):
              raise TypeError("other must be a free module")
  
          if self.ambient_vector_space() != other.ambient_vector_space():
              raise ArithmeticError("self and other must be embedded in the same ambient space.")
  
          if self.base_ring() != other.base_ring():
              if other.base_ring().is_field():
                  return other.intersection(self)
              raise NotImplementedError("intersection of modules over different base rings (neither a field) is not implemented.")
  
          # dispense with the three easy cases
          if self == self.ambient_vector_space() or other.is_submodule(self):
              return other
          elif other == other.ambient_vector_space() or self.is_submodule(other):
              return self
          elif self.rank() == 0 or other.rank() == 0:
              if self.base_ring().is_field():
                  return other.zero_submodule()
              else:
                  return self.zero_submodule()
  
          # standard algorithm for computing intersection of general submodule
          if self.dimension() <= other.dimension():
              V1 = self
              V2 = other
          else:
              V1 = other
              V2 = self
          A1 = V1.basis_matrix()
          A2 = V2.basis_matrix()
          S  = A1.stack(A2)
          K  = S.integer_kernel(self.base_ring()).basis_matrix()
          n  = int(V1.dimension())
          K = K.matrix_from_columns(range(n))
          B = K*A1
          return self.span(B)
  
      def __and__(self, other):
          r"""
          Return the intersection of ``self`` and ``other``.
  
          See :meth:`intersection`.
  
          EXAMPLES:
  
          We intersect two submodules one of which is clearly
          contained in the other::
  
              sage: A = ZZ^2
              sage: M1 = A.span([[1,1]])
              sage: M2 = A.span([[3,3]])
              sage: M1 & M2
              Free module of degree 2 and rank 1 over Integer Ring
              Echelon basis matrix:
              [3 3]
              sage: M1 & M2 is M2
              True
  
          We intersection two submodules of `\ZZ^3` of rank
          `2`, whose intersection has rank `1`::
  
              sage: A = ZZ^3
              sage: M1 = A.span([[1,1,1], [1,2,3]])
              sage: M2 = A.span([[2,2,2], [1,0,0]])
              sage: M1 & M2
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [2 2 2]
          """
          return self.intersection(other)
  
      def zero_submodule(self):
          """
          Return the zero submodule of this module.
  
          EXAMPLES::
  
              sage: V = FreeModule(ZZ,2)
              sage: V.zero_submodule()
              Free module of degree 2 and rank 0 over Integer Ring
              Echelon basis matrix:
              []
          """
          return self.submodule([], check=False, already_echelonized=True)
  
      def denominator(self):
          """
          The denominator of the basis matrix of ``self`` (i.e. the LCM of the
          coordinate entries with respect to the basis of the ambient
          space).
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: L = V.span([[1,1/2,1/3], [-1/5,2/3,3]],ZZ)
              sage: L
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 1/5 19/6 37/3]
              [   0 23/6 46/3]
              sage: L.denominator()
              30
          """
          return self.basis_matrix().denominator()
  
      def index_in_saturation(self):
          r"""
          Return the index of this module in its saturation, i.e., its
          intersection with `R^n`.
  
          EXAMPLES::
  
              sage: W = span([[2,4,6]], ZZ)
              sage: W.index_in_saturation()
              2
              sage: W = span([[1/2,1/3]], ZZ)
              sage: W.index_in_saturation()
              1/6
          """
          # TODO: There is probably a much faster algorithm in this case.
          return self.index_in(self.saturation())
  
      def saturation(self):
          r"""
          Return the saturated submodule of `R^n` that spans the same
          vector space as self.
  
          EXAMPLES:
  
          We create a 1-dimensional lattice that is obviously not
          saturated and saturate it.
  
          ::
  
              sage: L = span([[9,9,6]], ZZ); L
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [9 9 6]
              sage: L.saturation()
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [3 3 2]
  
          We create a lattice spanned by two vectors, and saturate.
          Computation of discriminants shows that the index of lattice in its
          saturation is `3`, which is a prime of congruence between
          the two generating vectors.
  
          ::
  
              sage: L = span([[1,2,3], [4,5,6]], ZZ)
              sage: L.saturation()
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              sage: L.discriminant()
              54
              sage: L.saturation().discriminant()
              6
  
          Notice that the saturation of a non-integral lattice `L` is
          defined, but the result is integral hence does not contain
          `L`::
  
              sage: L = span([['1/2',1,3]], ZZ)
              sage: L.saturation()
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [1 2 6]
  
          TESTS:
  
          We check that :trac:`24702` is fixed::
  
              sage: L = FreeQuadraticModule(ZZ,1,matrix.identity(1))
              sage: S = 2*L
              sage: S.saturation().ambient_module() == L
              True
          """
          R = self.base_ring()
          if R.is_field():
              return self
          try:
              A, _ = self.basis_matrix()._clear_denom()
              S = self.span(A.saturation())
          except AttributeError:
              # fallback in case _clear_denom isn't written
              V = self.vector_space()
              A = self.ambient_module()
              S = V.intersection(A)
          # Return exactly self if it is already saturated.
          return self if self == S else S
  
      def span(self, gens, base_ring=None, check=True, already_echelonized=False):
          """
          Return the R-span of the given list of gens, where R = base_ring.
          The default R is the base ring of self. Note that this span need
          not be a submodule of self, nor even of the ambient space. It must,
          however, be contained in the ambient vector space, i.e., the
          ambient space tensored with the fraction field of R.
  
          EXAMPLES::
  
              sage: V = FreeModule(ZZ,3)
              sage: W = V.submodule([V.gen(0)])
              sage: W.span([V.gen(1)])
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [0 1 0]
              sage: W.submodule([V.gen(1)])
              Traceback (most recent call last):
              ...
              ArithmeticError: Argument gens (= [(0, 1, 0)]) does not generate a submodule of self.
              sage: V.span([[1,0,0],[1/5,4,0],[6,3/4,0]])
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [1/5   0   0]
              [  0 1/4   0]
  
          It also works with other things than integers::
  
              sage: R.<x>=QQ[]
              sage: L=R^1
              sage: a=L.span([(1/x,)])
              sage: a
              Free module of degree 1 and rank 1 over Univariate Polynomial Ring in x over Rational Field
              Echelon basis matrix:
              [1/x]
              sage: b=L.span([(1/x,)])
              sage: a(b.gens()[0])
              (1/x)
              sage: L2 = R^2
              sage: L2.span([[(x^2+x)/(x^2-3*x+2),1/5],[(x^2+2*x)/(x^2-4*x+3),x]])
              Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field
              Echelon basis matrix:
              [x/(x^3 - 6*x^2 + 11*x - 6)  2/15*x^2 - 17/75*x - 1/75]
              [                         0 x^3 - 11/5*x^2 - 3*x + 4/5]
  
          Note that the ``base_ring`` can make a huge difference. We
          repeat the previous example over the fraction field of R and
          get a simpler vector space. ::
  
              sage: L2.span([[(x^2+x)/(x^2-3*x+2),1/5],[(x^2+2*x)/(x^2-4*x+3),x]],base_ring=R.fraction_field())
              Vector space of degree 2 and dimension 2 over Fraction Field of Univariate Polynomial Ring in x over Rational Field
              Basis matrix:
              [1 0]
              [0 1]
          """
          if is_FreeModule(gens):
              gens = gens.gens()
          if base_ring is None or base_ring is self.base_ring():
              return FreeModule_submodule_pid(
                  self.ambient_module(), gens, check=check, already_echelonized=already_echelonized)
          else:
              try:
                  M = self.change_ring(base_ring)
              except TypeError:
                  raise ValueError("Argument base_ring (= %s) is not compatible "%base_ring + \
                      "with the base field (= %s)." % self.base_field())
              try:
                  return M.span(gens)
              except TypeError:
                  raise ValueError("Argument gens (= %s) is not compatible "%gens + \
                      "with base_ring (= %s)."%base_ring)
  
      def submodule(self, gens, check=True, already_echelonized=False):
          r"""
          Create the R-submodule of the ambient vector space with given
          generators, where R is the base ring of self.
  
          INPUT:
  
  
          -  ``gens`` - a list of free module elements or a free
             module
  
          -  ``check`` - (default: True) whether or not to verify
             that the gens are in self.
  
  
          OUTPUT:
  
  
          -  ``FreeModule`` - the submodule spanned by the
             vectors in the list gens. The basis for the subspace is always put
             in reduced row echelon form.
  
  
          EXAMPLES:
  
          We create a submodule of `\ZZ^3`::
  
              sage: M = FreeModule(ZZ, 3)
              sage: B = M.basis()
              sage: W = M.submodule([B[0]+B[1], 2*B[1]-B[2]])
              sage: W
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 1  1  0]
              [ 0  2 -1]
  
          We create a submodule of a submodule.
  
          ::
  
              sage: W.submodule([3*B[0] + 3*B[1]])
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [3 3 0]
  
          We try to create a submodule that isn't really a submodule,
          which results in an ``ArithmeticError`` exception::
  
              sage: W.submodule([B[0] - B[1]])
              Traceback (most recent call last):
              ...
              ArithmeticError: Argument gens (= [(1, -1, 0)]) does not generate a submodule of self.
  
          Next we create a submodule of a free module over the principal ideal
          domain `\QQ[x]`, which uses the general Hermite normal form functionality::
  
              sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
              sage: M = FreeModule(R, 3)
              sage: B = M.basis()
              sage: W = M.submodule([x*B[0], 2*B[1]- x*B[2]]); W
              Free module of degree 3 and rank 2 over Univariate Polynomial Ring in x over Rational Field
              Echelon basis matrix:
              [ x  0  0]
              [ 0  2 -x]
              sage: W.ambient_module()
              Ambient free module of rank 3 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
          """
          if is_FreeModule(gens):
              gens = gens.gens()
          V = self.span(gens, check=check, already_echelonized=already_echelonized)
          if check:
              if not V.is_submodule(self):
                  raise ArithmeticError("Argument gens (= %s) does not generate a submodule of self."%gens)
          return V
  
      def span_of_basis(self, basis, base_ring=None, check=True, already_echelonized=False):
          r"""
          Return the free R-module with the given basis, where R is the base
          ring of ``self`` or user specified base_ring.
  
          Note that this R-module need not be a submodule of self, nor even
          of the ambient space. It must, however, be contained in the ambient
          vector space, i.e., the ambient space tensored with the fraction
          field of R.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ,3)
              sage: W = M.span_of_basis([M([1,2,3])])
  
          Next we create two free `\ZZ`-modules, neither of
          which is a submodule of `W`.
  
          ::
  
              sage: W.span_of_basis([M([2,4,0])])
              Free module of degree 3 and rank 1 over Integer Ring
              User basis matrix:
              [2 4 0]
  
          The following module isn't in the ambient module `\ZZ^3`
          but is contained in the ambient vector space `\QQ^3`::
  
              sage: V = M.ambient_vector_space()
              sage: W.span_of_basis([ V([1/5,2/5,0]), V([1/7,1/7,0]) ])
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [1/5 2/5   0]
              [1/7 1/7   0]
  
          Of course the input basis vectors must be linearly independent::
  
              sage: W.span_of_basis([ [1,2,0], [2,4,0] ])
              Traceback (most recent call last):
              ...
              ValueError: The given basis vectors must be linearly independent.
          """
          if is_FreeModule(basis):
              basis = basis.gens()
          if base_ring is None or base_ring == self.base_ring():
              try:
                  if self.is_dense():
                      from .free_module_integer import FreeModule_submodule_with_basis_integer
                      return FreeModule_submodule_with_basis_integer(self.ambient_module(),
                                                                     basis=basis, check=check,
                                                                     already_echelonized=already_echelonized,
                                                                     lll_reduce=False)
              except TypeError:
                  pass
  
              return FreeModule_submodule_with_basis_pid(
                  self.ambient_module(), basis=basis, check=check,
                  already_echelonized=already_echelonized)
          else:
              try:
                  M = self.change_ring(base_ring)
              except TypeError:
                  raise ValueError("Argument base_ring (= %s) is not compatible "%base_ring + \
                      "with the base ring (= %s)."%self.base_ring())
              try:
                  return M.span_of_basis(basis)
              except TypeError:
                  raise ValueError("Argument gens (= %s) is not compatible "%basis + \
                      "with base_ring (= %s)."%base_ring)
  
      def submodule_with_basis(self, basis, check=True, already_echelonized=False):
          r"""
          Create the R-submodule of the ambient vector space with given
          basis, where R is the base ring of self.
  
          INPUT:
  
          - ``basis`` -- a list of linearly independent vectors
  
          - ``check`` -- whether or not to verify that each gen is in
             the ambient vector space
  
          OUTPUT:
  
          - ``FreeModule`` -- the `R`-submodule with given basis
  
          EXAMPLES:
  
          First we create a submodule of `\\ZZ^3`::
  
              sage: M = FreeModule(ZZ, 3)
              sage: B = M.basis()
              sage: N = M.submodule_with_basis([B[0]+B[1], 2*B[1]-B[2]])
              sage: N
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [ 1  1  0]
              [ 0  2 -1]
  
          A list of vectors in the ambient vector space may fail to generate
          a submodule.
  
          ::
  
              sage: V = M.ambient_vector_space()
              sage: X = M.submodule_with_basis([ V(B[0]+B[1])/2, V(B[1]-B[2])/2])
              Traceback (most recent call last):
              ...
              ArithmeticError: The given basis does not generate a submodule of self.
  
          However, we can still determine the R-span of vectors in the
          ambient space, or over-ride the submodule check by setting check to
          False.
  
          ::
  
              sage: X = V.span([ V(B[0]+B[1])/2, V(B[1]-B[2])/2 ], ZZ)
              sage: X
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 1/2    0   1/2]
              [   0  1/2  -1/2]
              sage: Y = M.submodule([ V(B[0]+B[1])/2, V(B[1]-B[2])/2 ], check=False)
              sage: X == Y
              True
  
          Next we try to create a submodule of a free module over the
          principal ideal domain `\QQ[x]`, using our general Hermite normal form implementation::
  
              sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
              sage: M = FreeModule(R, 3)
              sage: B = M.basis()
              sage: W = M.submodule_with_basis([x*B[0], 2*B[0]- x*B[2]]); W
              Free module of degree 3 and rank 2 over Univariate Polynomial Ring in x over Rational Field
              User basis matrix:
              [ x  0  0]
              [ 2  0 -x]
          """
          V = self.span_of_basis(basis=basis, check=check, already_echelonized=already_echelonized)
          if check:
              if not V.is_submodule(self):
                  raise ArithmeticError("The given basis does not generate a submodule of self.")
          return V
  
      def vector_space_span(self, gens, check=True):
          r"""
          Create the vector subspace of the ambient vector space with given
          generators.
  
          INPUT:
  
  
          -  ``gens`` - a list of vector in self
  
          -  ``check`` - whether or not to verify that each gen
             is in the ambient vector space
  
  
          OUTPUT: a vector subspace
  
          EXAMPLES:
  
          We create a `2`-dimensional subspace of `\QQ^3`.
  
          ::
  
              sage: V = VectorSpace(QQ, 3)
              sage: B = V.basis()
              sage: W = V.vector_space_span([B[0]+B[1], 2*B[1]-B[2]])
              sage: W
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [   1    0  1/2]
              [   0    1 -1/2]
  
          We create a subspace of a vector space over
          `\QQ(i)`.
  
          ::
  
              sage: R.<x> = QQ[]
              sage: K = NumberField(x^2 + 1, 'a'); a = K.gen()
              sage: V = VectorSpace(K, 3)
              sage: V.vector_space_span([2*V.gen(0) + 3*V.gen(2)])
              Vector space of degree 3 and dimension 1 over Number Field in a with defining polynomial x^2 + 1
              Basis matrix:
              [  1   0 3/2]
  
          We use the ``vector_space_span`` command to create a
          vector subspace of the ambient vector space of a submodule of
          `\ZZ^3`.
  
          ::
  
              sage: M = FreeModule(ZZ,3)
              sage: W = M.submodule([M([1,2,3])])
              sage: W.vector_space_span([M([2,3,4])])
              Vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [  1 3/2   2]
          """
          if is_FreeModule(gens):
              gens = gens.gens()
          return FreeModule_submodule_field(self.ambient_vector_space(), gens, check=check)
  
      def vector_space_span_of_basis(self, basis, check=True):
          """
          Create the vector subspace of the ambient vector space with given
          basis.
  
          INPUT:
  
          - ``basis`` -- a list of linearly independent vectors
  
          - ``check`` -- whether or not to verify that each gen is in
             the ambient vector space
  
          OUTPUT: a vector subspace with user-specified basis
  
          EXAMPLES::
  
              sage: V = VectorSpace(QQ, 3)
              sage: B = V.basis()
              sage: W = V.vector_space_span_of_basis([B[0]+B[1], 2*B[1]-B[2]])
              sage: W
              Vector space of degree 3 and dimension 2 over Rational Field
              User basis matrix:
              [ 1  1  0]
              [ 0  2 -1]
          """
          return FreeModule_submodule_with_basis_field(self.ambient_vector_space(), basis, check=check)
  
      def quotient(self, sub, check=True, **kwds):
          """
          Return the quotient of ``self`` by the given submodule sub.
  
          INPUT:
  
          -  ``sub`` - a submodule of self, or something that can
             be turned into one via self.submodule(sub).
  
          -  ``check`` - (default: True) whether or not to check
             that sub is a submodule.
  
          -  further named arguments, that are passed to the constructor
             of the quotient space.
  
  
          EXAMPLES::
  
              sage: A = ZZ^3; V = A.span([[1,2,3], [4,5,6]])
              sage: Q = V.quotient( [V.0 + V.1] ); Q
              Finitely generated module V/W over Integer Ring with invariants (0)
          """
          # Calling is_subspace may be way too slow and repeat work done below.
          # It will be very desirable to somehow do this step better.
          if check and (not is_FreeModule(sub) or not sub.is_submodule(self)):
              try:
                  sub = self.submodule(sub)
              except (TypeError, ArithmeticError):
                  raise ArithmeticError("sub must be a subspace of self")
          if self.base_ring() == sage.rings.integer_ring.ZZ:
              from .fg_pid.fgp_module import FGP_Module
              return FGP_Module(self, sub, check=False, **kwds)
          else:
              raise NotImplementedError("quotients of modules over rings other than fields or ZZ is not fully implemented")
  
      def __truediv__(self, sub):
          """
          Return the quotient of ``self`` by the given submodule sub.
  
          EXAMPLES::
  
              sage: V1 = ZZ^2; W1 = V1.span([[1,2],[3,4]])
              sage: V1/W1
              Finitely generated module V/W over Integer Ring with invariants (2)
              sage: V2 = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W2 = V2.span([2*V2.0+4*V2.1, 9*V2.0+12*V2.1, 4*V2.2])
              sage: V2/W2
              Finitely generated module V/W over Integer Ring with invariants (4, 12)
          """
          return self.quotient(sub, check=True)
  
  class FreeModule_generic_field(FreeModule_generic_pid):
      """
      Base class for all free modules over fields.
      """
      def __init__(self, base_field, dimension, degree, sparse=False):
          """
          Creates a vector space over a field.
  
          EXAMPLES::
  
              sage: FreeModule(QQ, 2)
              Vector space of dimension 2 over Rational Field
              sage: FreeModule(FiniteField(2), 7)
              Vector space of dimension 7 over Finite Field of size 2
  
          We test that objects of this type are initialised correctly;
          see :trac:`11166` (the failing ``repr`` is fine because this
          is an abstract base class)::
  
              sage: from sage.modules.free_module import FreeModule_generic_field
              sage: FreeModule_generic_field(QQ, 5, 5)
              <repr(<sage.modules.free_module.FreeModule_generic_field_with_category at 0x...>) failed: NotImplementedError>
          """
          if not isinstance(base_field, ring.Field):
              raise TypeError("The base_field (=%s) must be a field"%base_field)
          FreeModule_generic_pid.__init__(self, base_field, dimension, degree, sparse=sparse)
  
      def _Hom_(self, Y, category):
          r"""
          Returns a homspace whose morphisms have this vector space as domain.
  
          This is called by the general methods such as
          :meth:`sage.structure.parent.Parent.Hom`.
  
          INPUT:
  
          - ``Y`` - a free module (or vector space) that will
            be the codomain of the morphisms in returned homspace
          - ``category`` - the category for the homspace
  
          OUTPUT:
  
          If ``Y`` is a free module over a field, in other words, a vector space,
          then this returns a space of homomorphisms between vector spaces,
          in other words a space of linear transformations.
  
          If ``Y`` is a free module that is not a vector space, then
          the returned space contains homomorphisms between free modules.
  
          EXAMPLES::
  
              sage: V = QQ^2
              sage: W = QQ^3
              sage: H = V._Hom_(W, category=None)
              sage: type(H)
              <class 'sage.modules.vector_space_homspace.VectorSpaceHomspace_with_category'>
              sage: H
              Set of Morphisms (Linear Transformations) from Vector space of dimension 2 over Rational Field to Vector space of dimension 3 over Rational Field
  
              sage: V = QQ^2
              sage: W = ZZ^3
              sage: H = V._Hom_(W, category=None)
              sage: type(H)
              <class 'sage.modules.free_module_homspace.FreeModuleHomspace_with_category'>
              sage: H
              Set of Morphisms from Vector space of dimension 2 over Rational Field
               to Ambient free module of rank 3 over the principal ideal domain Integer Ring
               in Category of finite dimensional vector spaces with basis over
                (number fields and quotient fields and metric spaces)
          """
          if Y.base_ring().is_field():
              from . import vector_space_homspace
              return vector_space_homspace.VectorSpaceHomspace(self, Y, category)
          from . import free_module_homspace
          return free_module_homspace.FreeModuleHomspace(self, Y, category)
  
      def scale(self, other):
          """
          Return the product of ``self`` by the number other, which is the module
          spanned by ``other`` times each basis vector. Since ``self`` is a vector
          space this product equals ``self`` if ``other`` is nonzero, and is the zero
          vector space if ``other`` is 0.
  
          EXAMPLES::
  
              sage: V = QQ^4
              sage: V.scale(5)
              Vector space of dimension 4 over Rational Field
              sage: V.scale(0)
              Vector space of degree 4 and dimension 0 over Rational Field
              Basis matrix:
              []
  
          ::
  
              sage: W = V.span([[1,1,1,1]])
              sage: W.scale(2)
              Vector space of degree 4 and dimension 1 over Rational Field
              Basis matrix:
              [1 1 1 1]
              sage: W.scale(0)
              Vector space of degree 4 and dimension 0 over Rational Field
              Basis matrix:
              []
  
          ::
  
              sage: V = QQ^4; V
              Vector space of dimension 4 over Rational Field
              sage: V.scale(3)
              Vector space of dimension 4 over Rational Field
              sage: V.scale(0)
              Vector space of degree 4 and dimension 0 over Rational Field
              Basis matrix:
              []
          """
          if other == 0:
              return self.zero_submodule()
          return self
  
      def __add__(self, other):
          """
          Return the sum of ``self`` and other.
  
          EXAMPLES::
  
              sage: V = VectorSpace(QQ,3)
              sage: V0 = V.span([V.gen(0)])
              sage: V2 = V.span([V.gen(2)])
              sage: V0 + V2
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [1 0 0]
              [0 0 1]
              sage: QQ^3 + 0
              Vector space of dimension 3 over Rational Field
          """
          if not isinstance(other, FreeModule_generic_field):
              if other == 0:
                  return self
              raise TypeError("other must be a Vector Space")
          V = self.ambient_vector_space()
          if V != other.ambient_vector_space():
              raise ArithmeticError("self and other must have the same ambient space")
          return V.span(self.basis() + other.basis())
  
      def echelonized_basis_matrix(self):
          """
          Return basis matrix for ``self`` in row echelon form.
  
          EXAMPLES::
  
              sage: V = FreeModule(QQ, 3).span_of_basis([[1,2,3],[4,5,6]])
              sage: V.basis_matrix()
              [1 2 3]
              [4 5 6]
              sage: V.echelonized_basis_matrix()
              [ 1  0 -1]
              [ 0  1  2]
          """
          return self.basis_matrix().echelon_form()
  
      def intersection(self, other):
          """
          Return the intersection of ``self`` and other, which must be
          R-submodules of a common ambient vector space.
  
          EXAMPLES::
  
              sage: V  = VectorSpace(QQ,3)
              sage: W1 = V.submodule([V.gen(0), V.gen(0) + V.gen(1)])
              sage: W2 = V.submodule([V.gen(1), V.gen(2)])
              sage: W1.intersection(W2)
              Vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [0 1 0]
              sage: W2.intersection(W1)
              Vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [0 1 0]
              sage: V.intersection(W1)
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [1 0 0]
              [0 1 0]
              sage: W1.intersection(V)
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [1 0 0]
              [0 1 0]
              sage: Z = V.submodule([])
              sage: W1.intersection(Z)
              Vector space of degree 3 and dimension 0 over Rational Field
              Basis matrix:
              []
          """
          if not isinstance(other, FreeModule_generic):
              raise TypeError("other must be a free module")
  
          if self.ambient_vector_space() != other.ambient_vector_space():
              raise ArithmeticError("self and other must have the same ambient space.")
  
          if self.rank() == 0 or other.rank() == 0:
              if self.base_ring().is_field():
                  return other.zero_submodule()
              else:
                  return self.zero_submodule()
  
          if self.base_ring() != other.base_ring():
              # Now other is over a ring R whose fraction field K is the base field of V = self.
              # We compute the intersection using the following algorithm:
              # 1. By explicitly computing the nullspace of the matrix whose rows
              #    are a basis for self, we obtain the matrix over a linear map
              #         phi:  K^n ----> W
              #    with kernel equal to V = self.
              # 2. Compute the kernel over R of Phi restricted to other.  Do this
              #    by clearing denominators, computing the kernel of a matrix with
              #    entries in R, then restoring denominators to the answer.
              K = self.base_ring()
              B = self.basis_matrix().transpose()
              W = B.kernel()
              phi = W.basis_matrix().transpose()
  
              # To restrict phi to other, we multiply the basis matrix for other
              # by phi, thus computing the image of each basis vector.
              X = other.basis_matrix()
              psi = X * phi
  
              # Now psi is a matrix that defines an R-module morphism from other to some
              # R-module, whose kernel defines the long sought for intersection of self and other.
              L = psi.integer_kernel()
  
              # Finally the kernel of the intersection has basis the linear combinations of
              # the basis of other given by a basis for L.
              G = L.basis_matrix() * other.basis_matrix()
              return other.span(G.rows())
  
          # dispense with the three easy cases
          if self == self.ambient_vector_space():
              return other
          elif other == other.ambient_vector_space():
              return self
          elif self.dimension() == 0 or other.dimension() == 0:
              return self.zero_submodule()
  
          # standard algorithm for computing intersection of general subspaces
          if self.dimension() <= other.dimension():
              V1 = self
              V2 = other
          else:
              V1 = other
              V2 = self
          A1 = V1.basis_matrix()
          A2 = V2.basis_matrix()
          S = A1.stack(A2)
          K = S.kernel()
          n = int(V1.dimension())
          B = [A1.linear_combination_of_rows(v.list()[:n]) for v in K.basis()]
          return self.ambient_vector_space().submodule(B, check=False)
  
      def is_subspace(self, other):
          """
          True if this vector space is a subspace of other.
  
          EXAMPLES::
  
              sage: V = VectorSpace(QQ,3)
              sage: W = V.subspace([V.gen(0), V.gen(0) + V.gen(1)])
              sage: W2 = V.subspace([V.gen(1)])
              sage: W.is_subspace(V)
              True
              sage: W2.is_subspace(V)
              True
              sage: W.is_subspace(W2)
              False
              sage: W2.is_subspace(W)
              True
          """
          return self.is_submodule(other)
  
      def span(self, gens, base_ring=None, check=True, already_echelonized=False):
          """
          Return the K-span of the given list of gens, where K is the
          base field of ``self`` or the user-specified base_ring.  Note that
          this span is a subspace of the ambient vector space, but need
          not be a subspace of self.
  
          INPUT:
  
  
          -  ``gens`` - list of vectors
  
          -  ``check`` - bool (default: True): whether or not to
             coerce entries of gens into base field
  
          -  ``already_echelonized`` - bool (default: False):
             set this if you know the gens are already in echelon form
  
  
          EXAMPLES::
  
              sage: V = VectorSpace(GF(7), 3)
              sage: W = V.subspace([[2,3,4]]); W
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [1 5 2]
              sage: W.span([[1,1,1]])
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [1 1 1]
  
          TESTS::
  
              sage: V = FreeModule(RDF,3)
              sage: W = V.submodule([V.gen(0)])
              sage: W.span([V.gen(1)], base_ring=GF(7))
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [0 1 0]
              sage: v = V((1, pi, log(2))); v
              (1.0, 3.141592653589793, 0.6931471805599453)
              sage: W.span([v], base_ring=GF(7))
              Traceback (most recent call last):
              ...
              ValueError: Argument gens (= [(1.0, 3.141592653589793, 0.6931471805599453)]) is not compatible with base_ring (= Finite Field of size 7).
              sage: W = V.submodule([v])
              sage: W.span([V.gen(2)], base_ring=GF(7))
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [0 0 1]
          """
          if is_FreeModule(gens):
              gens = gens.gens()
          if base_ring is None or base_ring is self.base_ring():
              return FreeModule_submodule_field(
                  self.ambient_module(), gens=gens, check=check, already_echelonized=already_echelonized)
          else:
              try:
                  M = self.ambient_module().change_ring(base_ring)
              except TypeError:
                  raise ValueError("Argument base_ring (= %s) is not compatible with the base field (= %s)." % (base_ring, self.base_field() ))
              try:
                  return M.span(gens)
              except TypeError:
                  raise ValueError("Argument gens (= %s) is not compatible with base_ring (= %s)." % (gens, base_ring))
  
      def span_of_basis(self, basis, base_ring=None, check=True, already_echelonized=False):
          r"""
          Return the free K-module with the given basis, where K is the base
          field of ``self`` or user specified base_ring.
  
          Note that this span is a subspace of the ambient vector space, but
          need not be a subspace of self.
  
          INPUT:
  
  
          -  ``basis`` - list of vectors
  
          -  ``check`` - bool (default: True): whether or not to
             coerce entries of gens into base field
  
          -  ``already_echelonized`` - bool (default: False):
             set this if you know the gens are already in echelon form
  
  
          EXAMPLES::
  
              sage: V = VectorSpace(GF(7), 3)
              sage: W = V.subspace([[2,3,4]]); W
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [1 5 2]
              sage: W.span_of_basis([[2,2,2], [3,3,0]])
              Vector space of degree 3 and dimension 2 over Finite Field of size 7
              User basis matrix:
              [2 2 2]
              [3 3 0]
  
          The basis vectors must be linearly independent or a
          ``ValueError`` exception is raised::
  
              sage: W.span_of_basis([[2,2,2], [3,3,3]])
              Traceback (most recent call last):
              ...
              ValueError: The given basis vectors must be linearly independent.
          """
          if is_FreeModule(basis):
              basis = basis.gens()
          if base_ring is None:
              return FreeModule_submodule_with_basis_field(
                  self.ambient_module(), basis=basis, check=check, already_echelonized=already_echelonized)
          else:
              try:
                  M = self.change_ring(base_ring)
              except TypeError:
                  raise ValueError("Argument base_ring (= %s) is not compatible with the base field (= %s)." % (
                      base_ring, self.base_field() ))
              try:
                  return M.span_of_basis(basis)
              except TypeError:
                  raise ValueError("Argument basis (= %s) is not compatible with base_ring (= %s)." % (basis, base_ring))
  
      def subspace(self, gens, check=True, already_echelonized=False):
          """
          Return the subspace of ``self`` spanned by the elements of gens.
  
          INPUT:
  
          -  ``gens`` - list of vectors
  
          -  ``check`` - bool (default: True) verify that gens
             are all in self.
  
          -  ``already_echelonized`` - bool (default: False) set
             to True if you know the gens are in Echelon form.
  
          EXAMPLES:
  
          First we create a 1-dimensional vector subspace of an
          ambient `3`-dimensional space over the finite field of
          order `7`::
  
              sage: V = VectorSpace(GF(7), 3)
              sage: W = V.subspace([[2,3,4]]); W
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [1 5 2]
  
          Next we create an invalid subspace, but it's allowed since
          ``check=False``. This is just equivalent to computing
          the span of the element::
  
              sage: W.subspace([[1,1,0]], check=False)
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [1 1 0]
  
          With ``check=True`` (the default) the mistake is correctly
          detected and reported with an ``ArithmeticError`` exception::
  
              sage: W.subspace([[1,1,0]], check=True)
              Traceback (most recent call last):
              ...
              ArithmeticError: Argument gens (= [[1, 1, 0]]) does not generate a submodule of self.
          """
          return self.submodule(gens, check=check, already_echelonized=already_echelonized)
  
      def subspaces(self, dim):
          """
          Iterate over all subspaces of dimension dim.
  
          INPUT:
  
          - ``dim`` - int, dimension of subspaces to be generated
  
          EXAMPLES::
  
              sage: V = VectorSpace(GF(3), 5)
              sage: len(list(V.subspaces(0)))
              1
              sage: len(list(V.subspaces(1)))
              121
              sage: len(list(V.subspaces(2)))
              1210
              sage: len(list(V.subspaces(3)))
              1210
              sage: len(list(V.subspaces(4)))
              121
              sage: len(list(V.subspaces(5)))
              1
  
          ::
  
              sage: V = VectorSpace(GF(3), 5)
              sage: V = V.subspace([V([1,1,0,0,0]),V([0,0,1,1,0])])
              sage: list(V.subspaces(1))
              [Vector space of degree 5 and dimension 1 over Finite Field of size 3
              Basis matrix:
              [1 1 0 0 0],
               Vector space of degree 5 and dimension 1 over Finite Field of size 3
              Basis matrix:
              [1 1 1 1 0],
               Vector space of degree 5 and dimension 1 over Finite Field of size 3
              Basis matrix:
              [1 1 2 2 0],
               Vector space of degree 5 and dimension 1 over Finite Field of size 3
              Basis matrix:
              [0 0 1 1 0]]
          """
          if not self.base_ring().is_finite():
              raise RuntimeError("Base ring must be finite.")
          b = self.basis_matrix()
          from sage.matrix.echelon_matrix import reduced_echelon_matrix_iterator
          for m in reduced_echelon_matrix_iterator(self.base_ring(), dim, self.dimension(), self.is_sparse(), copy=False):
              yield self.subspace((m*b).rows())
  
      def subspace_with_basis(self, gens, check=True, already_echelonized=False):
          """
          Same as ``self.submodule_with_basis(...)``.
  
          EXAMPLES:
  
          We create a subspace with a user-defined basis.
  
          ::
  
              sage: V = VectorSpace(GF(7), 3)
              sage: W = V.subspace_with_basis([[2,2,2], [1,2,3]]); W
              Vector space of degree 3 and dimension 2 over Finite Field of size 7
              User basis matrix:
              [2 2 2]
              [1 2 3]
  
          We then create a subspace of the subspace with user-defined basis.
  
          ::
  
              sage: W1 = W.subspace_with_basis([[3,4,5]]); W1
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              User basis matrix:
              [3 4 5]
  
          Notice how the basis for the same subspace is different if we
          merely use the ``subspace`` command.
  
          ::
  
              sage: W2 = W.subspace([[3,4,5]]); W2
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [1 6 4]
  
          Nonetheless the two subspaces are equal (as mathematical objects)::
  
              sage: W1 == W2
              True
          """
          return self.submodule_with_basis(gens, check=check, already_echelonized=already_echelonized)
  
      def complement(self):
          r"""
          Return the complement of ``self`` in the
          :meth:`~sage.modules.free_module.FreeModule_ambient_field.ambient_vector_space`.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: V.complement()
              Vector space of degree 3 and dimension 0 over Rational Field
              Basis matrix:
              []
              sage: V == V.complement().complement()
              True
              sage: W = V.span([[1, 0, 1]])
              sage: X = W.complement(); X
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  0]
              sage: X.complement() == W
              True
              sage: X + W == V
              True
  
          Even though we construct a subspace of a subspace, the
          orthogonal complement is still done in the ambient vector
          space `\QQ^3`::
  
              sage: V = QQ^3
              sage: W = V.subspace_with_basis([[1,0,1],[-1,1,0]])
              sage: X = W.subspace_with_basis([[1,0,1]])
              sage: X.complement()
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  0]
  
          All these complements are only done with respect to the inner
          product in the usual basis.  Over finite fields, this means
          we can get complements which are only isomorphic to a vector
          space decomposition complement. ::
  
              sage: F2 = GF(2,x)
              sage: V = F2^6
              sage: W = V.span([[1,1,0,0,0,0]])
              sage: W
              Vector space of degree 6 and dimension 1 over Finite Field of size 2
              Basis matrix:
              [1 1 0 0 0 0]
              sage: W.complement()
              Vector space of degree 6 and dimension 5 over Finite Field of size 2
              Basis matrix:
              [1 1 0 0 0 0]
              [0 0 1 0 0 0]
              [0 0 0 1 0 0]
              [0 0 0 0 1 0]
              [0 0 0 0 0 1]
              sage: W.intersection(W.complement())
              Vector space of degree 6 and dimension 1 over Finite Field of size 2
              Basis matrix:
              [1 1 0 0 0 0]
          """
          # Check simple cases
          if self.dimension() == 0:
              return self.ambient_vector_space()
          if self.dimension() == self.ambient_vector_space().dimension():
              return self.submodule([])
          return self.basis_matrix().right_kernel()
  
      def vector_space(self, base_field=None):
          """
          Return the vector space associated to self. Since ``self`` is a vector
          space this function simply returns self, unless the base field is
          different.
  
          EXAMPLES::
  
              sage: V = span([[1,2,3]],QQ); V
              Vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [1 2 3]
              sage: V.vector_space()
              Vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [1 2 3]
          """
          if base_field is None:
              return self
          return self.change_ring(base_field)
  
      def zero_submodule(self):
          """
          Return the zero submodule of self.
  
          EXAMPLES::
  
              sage: (QQ^4).zero_submodule()
              Vector space of degree 4 and dimension 0 over Rational Field
              Basis matrix:
              []
          """
          return self.zero_subspace()
  
      def zero_subspace(self):
          """
          Return the zero subspace of self.
  
          EXAMPLES::
  
              sage: (QQ^4).zero_subspace()
              Vector space of degree 4 and dimension 0 over Rational Field
              Basis matrix:
              []
          """
          return self.submodule([], check=False, already_echelonized=True)
  
      def linear_dependence(self, vectors, zeros='left', check=True):
          r"""
          Returns a list of vectors giving relations of linear dependence for the input list of vectors.
          Can be used to check linear independence of a set of vectors.
  
          INPUT:
  
          - ``vectors`` -- A list of vectors, all from the same vector
            space.
  
          - ``zeros`` -- default: ``'left'`` - ``'left'`` or ``'right'``
            as a general preference for where zeros are located in the
            returned coefficients
  
          - ``check`` -- default: ``True`` - if ``True`` each item in
            the list ``vectors`` is checked for membership in ``self``.
            Set to ``False`` if you can be certain the vectors come from
            the vector space.
  
          OUTPUT:
  
          Returns a list of vectors.  The scalar entries of each vector provide
          the coefficients for a linear combination of the input vectors that
          will equal the zero vector in ``self``.  Furthermore, the returned list
          is linearly independent in the vector space over the same base field
          with degree equal to the length of the list ``vectors``.
  
          The linear independence of ``vectors`` is equivalent to the returned
          list being empty, so this provides a test - see the examples below.
  
          The returned vectors are always independent, and with ``zeros`` set to
          ``'left'`` they have 1's in their first non-zero entries and a qualitative
          disposition to having zeros in the low-index entries.  With ``zeros`` set
          to ``'right'`` the situation is reversed with a qualitative disposition
          for zeros in the high-index entries.
  
          If the vectors in ``vectors`` are made the rows of a matrix `V` and
          the returned vectors are made the rows of a matrix `R`, then the
          matrix product `RV` is a zero matrix of the proper size.  And
          `R` is a matrix of full rank.  This routine uses kernels of
          matrices to compute these relations of linear dependence,
          but handles all the conversions between sets of vectors
          and matrices. If speed is important, consider working with
          the appropriate matrices and kernels instead.
  
          EXAMPLES:
  
          We begin with two linearly independent vectors, and add three
          non-trivial linear combinations to the set.  We illustrate
          both types of output and check a selected relation of linear
          dependence. ::
  
              sage: v1 = vector(QQ, [2, 1, -4, 3])
              sage: v2 = vector(QQ, [1, 5, 2, -2])
              sage: V = QQ^4
              sage: V.linear_dependence([v1,v2])
              [
              <BLANKLINE>
              ]
  
              sage: v3 = v1 + v2
              sage: v4 = 3*v1 - 4*v2
              sage: v5 = -v1 + 2*v2
              sage: L = [v1, v2, v3, v4, v5]
  
              sage: relations = V.linear_dependence(L, zeros='left')
              sage: relations
              [
              (1, 0, 0, -1, -2),
              (0, 1, 0, -1/2, -3/2),
              (0, 0, 1, -3/2, -7/2)
              ]
              sage: v2 + (-1/2)*v4 + (-3/2)*v5
              (0, 0, 0, 0)
  
              sage: relations = V.linear_dependence(L, zeros='right')
              sage: relations
              [
              (-1, -1, 1, 0, 0),
              (-3, 4, 0, 1, 0),
              (1, -2, 0, 0, 1)
              ]
              sage: z = sum([relations[2][i]*L[i] for i in range(len(L))])
              sage: z == zero_vector(QQ, 4)
              True
  
          A linearly independent set returns an empty list,
          a result that can be tested. ::
  
              sage: v1 = vector(QQ, [0,1,-3])
              sage: v2 = vector(QQ, [4,1,0])
              sage: V = QQ^3
              sage: relations = V.linear_dependence([v1, v2]); relations
              [
              <BLANKLINE>
              ]
              sage: relations == []
              True
  
          Exact results result from exact fields. We start with three
          linearly independent vectors and add in two linear combinations
          to make a linearly dependent set of five vectors. ::
  
              sage: F = FiniteField(17)
              sage: v1 = vector(F, [1, 2, 3, 4, 5])
              sage: v2 = vector(F, [2, 4, 8, 16, 15])
              sage: v3 = vector(F, [1, 0, 0, 0, 1])
              sage: (F^5).linear_dependence([v1, v2, v3]) == []
              True
              sage: L = [v1, v2, v3, 2*v1+v2, 3*v2+6*v3]
              sage: (F^5).linear_dependence(L)
              [
              (1, 0, 16, 8, 3),
              (0, 1, 2, 0, 11)
              ]
              sage: v1 + 16*v3 + 8*(2*v1+v2) + 3*(3*v2+6*v3)
              (0, 0, 0, 0, 0)
              sage: v2 + 2*v3 + 11*(3*v2+6*v3)
              (0, 0, 0, 0, 0)
              sage: (F^5).linear_dependence(L, zeros='right')
              [
              (15, 16, 0, 1, 0),
              (0, 14, 11, 0, 1)
              ]
  
          TESTS:
  
          With ``check=True`` (the default) a mismatch between vectors
          and the vector space is caught. ::
  
              sage: v1 = vector(RR, [1,2,3])
              sage: v2 = vector(RR, [1,2,3,4])
              sage: (RR^3).linear_dependence([v1,v2], check=True)
              Traceback (most recent call last):
              ...
              ValueError: vector (1.00000000000000, 2.00000000000000, 3.00000000000000, 4.00000000000000) is not an element of Vector space of dimension 3 over Real Field with 53 bits of precision
  
          The ``zeros`` keyword is checked. ::
  
              sage: (QQ^3).linear_dependence([vector(QQ,[1,2,3])], zeros='bogus')
              Traceback (most recent call last):
              ...
              ValueError: 'zeros' keyword must be 'left' or 'right', not 'bogus'
  
          An empty input set is linearly independent, vacuously. ::
  
              sage: (QQ^3).linear_dependence([]) == []
              True
          """
          if check:
              for v in vectors:
                  if not v in self:
                      raise ValueError('vector %s is not an element of %s' % (v, self))
          if zeros == 'left':
              basis = 'echelon'
          elif zeros == 'right':
              basis = 'pivot'
          else:
              raise ValueError("'zeros' keyword must be 'left' or 'right', not '%s'" % zeros)
          import sage.matrix.constructor
          A = sage.matrix.constructor.matrix(vectors)  # as rows, so get left kernel
          return A.left_kernel(basis=basis).basis()
  
      def __truediv__(self, sub):
          """
          Return the quotient of ``self`` by the given subspace sub.
  
          EXAMPLES::
  
              sage: V = RDF^3; W = V.span([[1,0,-1], [1,-1,0]])
              sage: Q = V/W; Q
              Vector space quotient V/W of dimension 1 over Real Double Field where
              V: Vector space of dimension 3 over Real Double Field
              W: Vector space of degree 3 and dimension 2 over Real Double Field
              Basis matrix:
              [ 1.0  0.0 -1.0]
              [ 0.0  1.0 -1.0]
              sage: type(Q)
              <class 'sage.modules.quotient_module.FreeModule_ambient_field_quotient_with_category'>
              sage: V([1,2,3])
              (1.0, 2.0, 3.0)
              sage: Q == V.quotient(W)
              True
              sage: Q(W.0)
              (0.0)
          """
          return self.quotient(sub, check=True)
  
      def quotient(self, sub, check=True):
          """
          Return the quotient of ``self`` by the given subspace sub.
  
          INPUT:
  
  
          -  ``sub`` - a submodule of self, or something that can
             be turned into one via self.submodule(sub).
  
          -  ``check`` - (default: True) whether or not to check
             that sub is a submodule.
  
  
          EXAMPLES::
  
              sage: A = QQ^3; V = A.span([[1,2,3], [4,5,6]])
              sage: Q = V.quotient( [V.0 + V.1] ); Q
              Vector space quotient V/W of dimension 1 over Rational Field where
              V: Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              W: Vector space of degree 3 and dimension 1 over Rational Field
              Basis matrix:
              [1 1 1]
              sage: Q(V.0 + V.1)
              (0)
  
          We illustrate that the base rings must be the same::
  
              sage: (QQ^2)/(ZZ^2)
              Traceback (most recent call last):
              ...
              ValueError: base rings must be the same
          """
          # Calling is_submodule may be way too slow and repeat work done below.
          # It will be very desirable to somehow do this step better.
          if is_FreeModule(sub) and self.base_ring() != sub.base_ring():
              raise ValueError("base rings must be the same")
          if check and (not is_FreeModule(sub) or not sub.is_subspace(self)):
              try:
                  sub = self.subspace(sub)
              except (TypeError, ArithmeticError):
                  raise ArithmeticError("sub must be a subspace of self")
          A, L = self.__quotient_matrices(sub)
          from . import quotient_module
          return quotient_module.FreeModule_ambient_field_quotient(self, sub, A, L)
  
      def __quotient_matrices(self, sub):
          r"""
          This internal function is used by
          ``self.quotient(...)``.
  
          EXAMPLES::
  
              sage: V = QQ^3; W = V.span([[1,0,-1], [1,-1,0]])
              sage: A, L = V._FreeModule_generic_field__quotient_matrices(W)
              sage: A
              [1]
              [1]
              [1]
              sage: L
              [1 0 0]
  
          The quotient and lift maps are used to compute in the quotient and
          to lift::
  
              sage: Q = V/W
              sage: Q(W.0)
              (0)
              sage: Q.lift_map()(Q.0)
              (1, 0, 0)
              sage: Q(Q.lift_map()(Q.0))
              (1)
  
          An example in characteristic 5::
  
              sage: A = GF(5)^2; B = A.span([[1,3]]); A / B
              Vector space quotient V/W of dimension 1 over Finite Field of size 5 where
              V: Vector space of dimension 2 over Finite Field of size 5
              W: Vector space of degree 2 and dimension 1 over Finite Field of size 5
              Basis matrix:
              [1 3]
          """
          # 2. Find a basis C for a another submodule of self, so that
          #    B + C is a basis for self.
          # 3. Then the quotient map is:
          #     x |---> 'write in terms of basis for C and take the last m = #C-#B components.
          # 4. And a section of this map is:
          #     x |---> corresponding linear combination of entries of last m entries
          #    of the basis C.
  
          # Step 1: Find bases for spaces
          B = sub.basis_matrix()
          S = self.basis_matrix()
  
          n = self.dimension()
          m = n - sub.dimension()
  
          # Step 2: Extend basis B to a basis for self.
          # We do this by simply finding the pivot rows of the matrix
          # whose rows are a basis for sub concatenated with a basis for
          # self.
          C = B.stack(S).transpose()
          A = C.matrix_from_columns(C.pivots()).transpose()
  
          # Step 3: Compute quotient map
          # The quotient map is given by writing in terms of the above basis,
          # then taking the last #C columns
  
          # Compute the matrix D "change of basis from S to A"
          # that writes each element of the basis
          # for ``self`` in terms of the basis of rows of A, i.e.,
          # want to find D such that
          #                D * A = S
          # where D is a square n x n matrix.
          # Our algorithm is to note that D is determined if we just
          # replace both A and S by the submatrix got from their pivot
          # columns.
          P  = A.pivots()
          AA = A.matrix_from_columns(P)
          SS = S.matrix_from_columns(P)
          D  = SS * AA**(-1)
  
          # Compute the image of each basis vector for ``self`` under the
          # map "write an element of ``self`` in terms of the basis A" then
          # take the last n-m components.
          Q = D.matrix_from_columns(range(n - m, n))
  
          # Step 4. Section map
          # The lifting or section map
          Dinv = D**(-1)
          L = Dinv.matrix_from_rows(range(n - m, n))
  
          return Q, L
  
      def quotient_abstract(self, sub, check=True, **kwds):
          r"""
          Return an ambient free module isomorphic to the quotient space
          of ``self`` modulo ``sub``, together with maps from ``self`` to
          the quotient, and a lifting map in the other direction.
  
          Use ``self.quotient(sub)`` to obtain the quotient
          module as an object equipped with natural maps in both directions,
          and a canonical coercion.
  
          INPUT:
  
          -  ``sub`` -- a submodule of ``self`` or something that can
             be turned into one via ``self.submodule(sub)``
  
          -  ``check`` -- (default: ``True``) whether or not to check
             that sub is a submodule
  
          - further named arguments, that are currently ignored.
  
          OUTPUT:
  
          -  ``U`` -- the quotient as an abstract *ambient* free module
  
          -  ``pi`` -- projection map to the quotient
  
          -  ``lift`` -- lifting map back from quotient
  
  
          EXAMPLES::
  
              sage: V = GF(19)^3
              sage: W = V.span_of_basis([ [1,2,3], [1,0,1] ])
              sage: U,pi,lift = V.quotient_abstract(W)
              sage: pi(V.2)
              (18)
              sage: pi(V.0)
              (1)
              sage: pi(V.0 + V.2)
              (0)
  
          Another example involving a quotient of one subspace by another::
  
              sage: A = matrix(QQ,4,4,[0,1,0,0, 0,0,1,0, 0,0,0,1, 0,0,0,0])
              sage: V = (A^3).kernel()
              sage: W = A.kernel()
              sage: U, pi, lift = V.quotient_abstract(W)
              sage: [pi(v) == 0 for v in W.gens()]
              [True]
              sage: [pi(lift(b)) == b for b in U.basis()]
              [True, True]
          """
          # Calling is_subspace may be way too slow and repeat work done below.
          # It will be very desirable to somehow do this step better.
          if check and (not is_FreeModule(sub) or not sub.is_subspace(self)):
              try:
                  sub = self.subspace(sub)
              except (TypeError, ArithmeticError):
                  raise ArithmeticError("sub must be a subspace of self")
  
          A, L = self.__quotient_matrices(sub)
          quomap = self.hom(A)
          quo = quomap.codomain()
          liftmap = quo.Hom(self)(L)
  
          return quomap.codomain(), quomap, liftmap
  
  ###############################################################################
  #
  # Generic ambient free modules, i.e., of the form R^n for some commutative ring R.
  #
  ###############################################################################
  
  class FreeModule_ambient(FreeModule_generic):
      """
      Ambient free module over a commutative ring.
      """
      def __init__(self, base_ring, rank, sparse=False, coordinate_ring=None):
          """
          The free module of given rank over the given base_ring.
  
          INPUT:
  
          -  ``base_ring`` -- a commutative ring
  
          -  ``rank`` -- a non-negative integer
  
          -  ``sparse`` -- bool (default: False)
  
          - ``coordinate_ring`` -- a ring containing ``base_ring``
            (default: equal to ``base_ring``)
  
          EXAMPLES::
  
              sage: FreeModule(ZZ, 4)
              Ambient free module of rank 4 over the principal ideal domain Integer Ring
  
          TESTS:
  
          We check that the creation of a submodule does not trigger
          the construction of a basis of the ambient space. See :trac:`15953`::
  
              sage: F.<a> = GF(4)
              sage: V = VectorSpace(F, 1)
              sage: v = V.random_element()
              sage: _ = V.subspace([v])
              sage: hasattr(V, '_FreeModule_ambient__basis')
              False
              sage: _ = V.basis()
              sage: hasattr(V, '_FreeModule_ambient__basis')
              True
          """
          FreeModule_generic.__init__(self, base_ring, rank=rank,
                  degree=rank, sparse=sparse, coordinate_ring=coordinate_ring)
  
      def __hash__(self):
          """
          The hash is obtained from the rank and the base ring.
  
          .. TODO::
  
              Make pickling so that the hash is available early enough.
  
          EXAMPLES::
  
              sage: V = QQ^7
              sage: hash(V) == hash((V.rank(), V.base_ring()))
              True
          """
          try:
              return hash((self.rank(), self.base_ring()))
          except AttributeError:
              # This is a fallback because sometimes hash is called during object
              # reconstruction (unpickle), and the above fields haven't been
              # filled in yet.
              return 0
  
      def _coerce_map_from_(self, M):
          """
          Return a coercion map from `M` to ``self``, or None.
  
          TESTS:
  
          Make sure :trac:`10513` is fixed (no coercion from a quotient
          vector space to an isomorphic abstract vector space)::
  
              sage: M = QQ^3 / [[1,2,3]]
              sage: V = QQ^2
              sage: V.coerce_map_from(M)
  
          """
          if isinstance(M, FreeModule_ambient):
              from sage.modules.quotient_module import FreeModule_ambient_field_quotient
              if isinstance(M, FreeModule_ambient_field_quotient):
                  # No forgetful map.
                  return None
              elif (self.base_ring().has_coerce_map_from(M.base_ring())
                    and self.rank() == M.rank()):
                  # We could return M.hom(self.basis(), self), but the
                  # complexity of this is quadratic in space and time,
                  # since it constructs a matrix.
                  return True
          return super(FreeModule_ambient, self)._coerce_map_from_(M)
  
      def _dense_module(self):
          """
          Creates a dense module with the same defining data as self.
  
          N.B. This function is for internal use only! See dense_module for
          use.
  
          EXAMPLES::
  
              sage: M = FreeModule(Integers(8),3)
              sage: S = FreeModule(Integers(8),3, sparse=True)
              sage: M is S._dense_module()
              True
          """
          return FreeModule(base_ring=self.base_ring(), rank = self.rank(), sparse=False)
  
      def _sparse_module(self):
          """
          Creates a sparse module with the same defining data as self.
  
          N.B. This function is for internal use only! See sparse_module for
          use.
  
          EXAMPLES::
  
              sage: M = FreeModule(Integers(8),3)
              sage: S = FreeModule(Integers(8),3, sparse=True)
              sage: M._sparse_module() is S
              True
          """
          return FreeModule(base_ring=self.base_ring(), rank = self.rank(), sparse=True)
  
      def echelonized_basis_matrix(self):
          """
          The echelonized basis matrix of self.
  
          EXAMPLES::
  
              sage: V = ZZ^4
              sage: W = V.submodule([ V.gen(i)-V.gen(0) for i in range(1,4) ])
              sage: W.basis_matrix()
              [ 1  0  0 -1]
              [ 0  1  0 -1]
              [ 0  0  1 -1]
              sage: W.echelonized_basis_matrix()
              [ 1  0  0 -1]
              [ 0  1  0 -1]
              [ 0  0  1 -1]
              sage: U = V.submodule_with_basis([ V.gen(i)-V.gen(0) for i in range(1,4) ])
              sage: U.basis_matrix()
              [-1  1  0  0]
              [-1  0  1  0]
              [-1  0  0  1]
              sage: U.echelonized_basis_matrix()
              [ 1  0  0 -1]
              [ 0  1  0 -1]
              [ 0  0  1 -1]
          """
          return self.basis_matrix()
  
      def _echelon_matrix_richcmp(self, other, op):
          r"""
          Compare the free module ``self`` with ``other`.
  
          This compares modules by their ambient spaces, then by dimension,
          then in order by their echelon matrices. However, if
          ``other`` is a sub-module or is a quotient module then its
          total comparison method is used instead of generic comparison.
  
          EXAMPLES:
  
          We compare rank three free modules over the integers and
          rationals::
  
              sage: from sage.structure.richcmp import op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE
              sage: (QQ^3)._echelon_matrix_richcmp(CC^3, op_LT)
              True
              sage: (CC^3)._echelon_matrix_richcmp(QQ^3, op_LT)
              False
              sage: (CC^3)._echelon_matrix_richcmp(QQ^3, op_GT)
              True
  
          ::
  
              sage: from sage.structure.richcmp import op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE
              sage: Q = QQ; Z = ZZ
              sage: (Q^3)._echelon_matrix_richcmp(Z^3, op_GT)
              True
              sage: (Q^3)._echelon_matrix_richcmp(Z^3, op_LT)
              False
              sage: (Z^3)._echelon_matrix_richcmp(Q^3, op_LT)
              True
              sage: (Z^3)._echelon_matrix_richcmp(Q^3, op_GT)
              False
              sage: (Q^3)._echelon_matrix_richcmp(Z^3, op_EQ)
              False
              sage: (Q^3)._echelon_matrix_richcmp(Q^3, op_EQ)
              True
  
          Comparison with a sub-module::
  
              sage: from sage.structure.richcmp import op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE
              sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
              sage: V
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              sage: A = QQ^3
              sage: V._echelon_matrix_richcmp(A, op_LT)
              True
              sage: A._echelon_matrix_richcmp(V, op_LT)
              False
  
          Comparison with a quotient module (see :trac:`10513`)::
  
              sage: from sage.structure.richcmp import op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE
              sage: M = QQ^3 / [[1,2,3]]
              sage: V = QQ^2
              sage: V._echelon_matrix_richcmp(M, op_EQ)
              False
              sage: M._echelon_matrix_richcmp(V, op_EQ)
              False
          """
          if self is other:
              return rich_to_bool(op, 0)
  
          if not isinstance(other, FreeModule_generic):
              return NotImplemented
  
          from sage.modules.quotient_module import FreeModule_ambient_field_quotient
          if isinstance(other, FreeModule_ambient):
              if (isinstance(other, FreeModule_ambient_field_quotient) or
                      isinstance(self, FreeModule_ambient_field_quotient)):
                  return richcmp(self,other,op)
  
              lx = self.rank()
              rx = other.rank()
              if lx != rx:
                  return richcmp_not_equal(lx, rx, op)
  
              lx = self.base_ring()
              rx = other.base_ring()
              if lx == rx:
                  # We do not want to create an inner product matrix in memory if
                  # self and other use the dot product
                  if self._inner_product_is_dot_product() and other._inner_product_is_dot_product():
                      return rich_to_bool(op, 0)
                  else:
                      #this only affects free_quadratic_modules
                      lx = self.inner_product_matrix()
                      rx = other.inner_product_matrix()
                      return richcmp(lx,rx,op)
  
              try:
                  if lx.is_subring(rx):
                      return rich_to_bool(op, -1)
                  elif rx.is_subring(lx):
                      return rich_to_bool(op, 1)
              except NotImplementedError:
                  pass
              return richcmp_not_equal(lx, rx, op)
          else:
              # now other is not ambient or is a quotient;
              # it knows how to do the comparison.
              return other._echelon_matrix_richcmp(self, revop(op))
  
      def _repr_(self):
          """
          The printing representation of self.
  
          EXAMPLES::
  
              sage: R = ZZ.quo(12)
              sage: M = R^12
              sage: M
              Ambient free module of rank 12 over Ring of integers modulo 12
              sage: print(M._repr_())
              Ambient free module of rank 12 over Ring of integers modulo 12
  
          The system representation can be overwritten, but leaves _repr_
          unmodified.
  
          ::
  
              sage: M.rename('M')
              sage: M
              M
              sage: print(M._repr_())
              Ambient free module of rank 12 over Ring of integers modulo 12
  
          Sparse modules print this fact.
  
          ::
  
              sage: N = FreeModule(R,12,sparse=True)
              sage: N
              Ambient sparse free module of rank 12 over Ring of integers modulo 12
  
          (Now clean up again.)
  
          ::
  
              sage: M.reset_name()
              sage: M
              Ambient free module of rank 12 over Ring of integers modulo 12
          """
          if self.is_sparse():
              return "Ambient sparse free module of rank %s over %s"%(self.rank(), self.base_ring())
          else:
              return "Ambient free module of rank %s over %s"%(self.rank(), self.base_ring())
  
      def _latex_(self):
          r"""
          Return a latex representation of this ambient free module.
  
          EXAMPLES::
  
              sage: latex(QQ^3) # indirect doctest
              \Bold{Q}^{3}
  
          ::
  
              sage: A = GF(5)^20
              sage: latex(A)  # indirect doctest
              \Bold{F}_{5}^{20}
  
          ::
  
              sage: A = PolynomialRing(QQ,3,'x') ^ 20
              sage: latex(A) #indirect doctest
              (\Bold{Q}[x_{0}, x_{1}, x_{2}])^{20}
          """
          t = "%s" % latex.latex(self.base_ring())
          if t.find(" ") != -1:
              t = "(%s)" % t
          return "%s^{%s}" % (t, self.rank())
  
      def is_ambient(self):
          """
          Return ``True`` since this module is an ambient
          module.
  
          EXAMPLES::
  
              sage: A = QQ^5; A.is_ambient()
              True
              sage: A = (QQ^5).span([[1,2,3,4,5]]); A.is_ambient()
              False
          """
          return True
  
      def ambient_module(self):
          """
          Return ``self``, since ``self`` is ambient.
  
          EXAMPLES::
  
              sage: A = QQ^5; A.ambient_module()
              Vector space of dimension 5 over Rational Field
              sage: A = ZZ^5; A.ambient_module()
              Ambient free module of rank 5 over the principal ideal domain Integer Ring
          """
          return self
  
      def basis(self):
          """
          Return a basis for this ambient free module.
  
          OUTPUT:
  
  
          -  ``Sequence`` - an immutable sequence with universe
             this ambient free module
  
  
          EXAMPLES::
  
              sage: A = ZZ^3; B = A.basis(); B
              [
              (1, 0, 0),
              (0, 1, 0),
              (0, 0, 1)
              ]
              sage: B.universe()
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
          """
          try:
              return self.__basis
          except  AttributeError:
              ZERO = self(0)
              one = self.coordinate_ring().one()
              w = []
              for n in range(self.rank()):
                  v = ZERO.__copy__()
                  v.set(n, one)
                  w.append(v)
              self.__basis = basis_seq(self, w)
              return self.__basis
  
      def echelonized_basis(self):
          """
          Return a basis for this ambient free module in echelon form.
  
          EXAMPLES::
  
              sage: A = ZZ^3; A.echelonized_basis()
              [
              (1, 0, 0),
              (0, 1, 0),
              (0, 0, 1)
              ]
          """
          return self.basis()
  
      def change_ring(self, R):
          """
          Return the ambient free module over R of the same rank as self.
  
          EXAMPLES::
  
              sage: A = ZZ^3; A.change_ring(QQ)
              Vector space of dimension 3 over Rational Field
              sage: A = ZZ^3; A.change_ring(GF(5))
              Vector space of dimension 3 over Finite Field of size 5
  
          For ambient modules any change of rings is defined.
  
          ::
  
              sage: A = GF(5)**3; A.change_ring(QQ)
              Vector space of dimension 3 over Rational Field
          """
          if self.base_ring() is R:
              return self
          from .free_quadratic_module import is_FreeQuadraticModule
          if is_FreeQuadraticModule(self):
              return FreeModule(R, self.rank(), inner_product_matrix=self.inner_product_matrix())
          else:
              return FreeModule(R, self.rank())
  
  
      def linear_combination_of_basis(self, v):
          """
          Return the linear combination of the basis for ``self`` obtained from
          the elements of the list v.
  
          INPUT:
  
          - ``v`` - list
  
          EXAMPLES::
  
              sage: V = span([[1,2,3], [4,5,6]], ZZ)
              sage: V
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [1 2 3]
              [0 3 6]
              sage: V.linear_combination_of_basis([1,1])
              (1, 5, 9)
  
          This should raise an error if the resulting element is not in self::
  
              sage: W = span([[2,4]], ZZ)
              sage: W.linear_combination_of_basis([1/2])
              Traceback (most recent call last):
              ...
              TypeError: element [1, 2] is not in free module
          """
          return self(v)
  
      def coordinate_vector(self, v, check=True):
          """
          Write `v` in terms of the standard basis for ``self`` and
          return the resulting coefficients in a vector over the fraction
          field of the base ring.
  
          Returns a vector `c` such that if `B` is the basis for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: V = Integers(16)^3
              sage: v = V.coordinate_vector([1,5,9]); v
              (1, 5, 9)
              sage: v.parent()
              Ambient free module of rank 3 over Ring of integers modulo 16
          """
          return self(v)
  
      def echelon_coordinate_vector(self, v, check=True):
          r"""
          Same as ``self.coordinate_vector(v)``, since ``self`` is
          an ambient free module.
  
          INPUT:
  
  
          -  ``v`` - vector
  
          -  ``check`` - bool (default: True); if True, also
             verify that v is really in self.
  
  
          OUTPUT: list
  
          EXAMPLES::
  
              sage: V = QQ^4
              sage: v = V([-1/2,1/2,-1/2,1/2])
              sage: v
              (-1/2, 1/2, -1/2, 1/2)
              sage: V.coordinate_vector(v)
              (-1/2, 1/2, -1/2, 1/2)
              sage: V.echelon_coordinate_vector(v)
              (-1/2, 1/2, -1/2, 1/2)
              sage: W = V.submodule_with_basis([[1/2,1/2,1/2,1/2],[1,0,1,0]])
              sage: W.coordinate_vector(v)
              (1, -1)
              sage: W.echelon_coordinate_vector(v)
              (-1/2, 1/2)
          """
          return self.coordinate_vector(v, check=check)
  
      def echelon_coordinates(self, v, check=True):
          """
          Returns the coordinate vector of v in terms of the echelon basis
          for self.
  
          EXAMPLES::
  
              sage: U = VectorSpace(QQ,3)
              sage: [ U.coordinates(v) for v in U.basis() ]
              [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
              sage: [ U.echelon_coordinates(v) for v in U.basis() ]
              [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
              sage: V = U.submodule([[1,1,0],[0,1,1]])
              sage: V
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  1]
              sage: [ V.coordinates(v) for v in V.basis() ]
              [[1, 0], [0, 1]]
              sage: [ V.echelon_coordinates(v) for v in V.basis() ]
              [[1, 0], [0, 1]]
              sage: W = U.submodule_with_basis([[1,1,0],[0,1,1]])
              sage: W
              Vector space of degree 3 and dimension 2 over Rational Field
              User basis matrix:
              [1 1 0]
              [0 1 1]
              sage: [ W.coordinates(v) for v in W.basis() ]
              [[1, 0], [0, 1]]
              sage: [ W.echelon_coordinates(v) for v in W.basis() ]
              [[1, 1], [0, 1]]
          """
          return self.coordinates(v, check=check)
  
      def random_element(self, prob=1.0, *args, **kwds):
          """
          Returns a random element of self.
  
          INPUT:
  
  
          - ``prob`` - float. Each coefficient will be set to zero with
             probability `1-prob`. Otherwise coefficients will be chosen
             randomly from base ring (and may be zero).
  
          - ``*args, **kwds`` - passed on to random_element function of base
             ring.
  
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ, 3)
              sage: M.random_element().parent() is M
              True
  
          Passes extra positional or keyword arguments through::
  
              sage: all(i in range(5, 10) for i in M.random_element(1.0, 5, 10))
              True
  
          ::
  
              sage: M = FreeModule(ZZ, 16)
              sage: M.random_element().parent() is M
              True
  
              sage: def add_sample(**kwds):
              ....:     global total, zeros
              ....:     v = M.random_element(**kwds)
              ....:     total += M.rank()
              ....:     zeros += sum(i == 0 for i in v)
  
              sage: total = 0
              sage: zeros = 0
              sage: add_sample()
              sage: expected = 1/5
              sage: while abs(zeros/total - expected) > 0.01:
              ....:     add_sample()
  
              sage: total = 0
              sage: zeros = 0
              sage: add_sample(prob=0.3)
              sage: expected = 1/5 * 3/10 + 7/10
              sage: while abs(zeros/total - expected) > 0.01:
              ....:     add_sample(prob=0.3)
  
              sage: total = 0
              sage: zeros = 0
              sage: add_sample(prob=0.7)
              sage: expected = 1/5 * 7/10 + 3/10
              sage: while abs(zeros/total - expected) > 0.01:
              ....:     add_sample(prob=0.7)
          """
          rand = current_randstate().python_random().random
          R = self.base_ring()
          v = self(0)
          prob = float(prob)
          for i in range(self.rank()):
              if rand() <= prob:
                  v[i] = R.random_element(*args, **kwds)
          return v
  
      def gen(self, i=0):
          """
          Return the `i`-th generator for ``self``.
  
          Here `i` is between 0 and rank - 1, inclusive.
  
          INPUT:
  
          - `i` -- an integer (default 0)
  
          OUTPUT: `i`-th basis vector for ``self``.
  
          EXAMPLES::
  
              sage: n = 5
              sage: V = QQ^n
              sage: B = [V.gen(i) for i in range(n)]
              sage: B
              [(1, 0, 0, 0, 0),
              (0, 1, 0, 0, 0),
              (0, 0, 1, 0, 0),
              (0, 0, 0, 1, 0),
              (0, 0, 0, 0, 1)]
              sage: V.gens() == tuple(B)
              True
  
          TESTS::
  
              sage: (QQ^3).gen(4/3)
              Traceback (most recent call last):
              ...
              TypeError: unable to convert rational 4/3 to an integer
  
          Check that :trac:`10262` and :trac:`13304` are fixed
          (coercions involving :class:`FreeModule_ambient` used to take
          quadratic time and space in the rank of the module)::
  
              sage: vector([0]*50000)/1
              (0, 0, 0, ..., 0)
          """
          if i < 0 or i >= self.rank():
              raise ValueError("Generator %s not defined." % i)
          try:
              return self.__basis[i]
          except AttributeError:
              v = self(0)
              v[i] = self.base_ring().one()
              v.set_immutable()
              return v
  
      def _sympy_(self):
          """
          Return a SymPy ``ProductSet`` corresponding to ``self``.
  
          EXAMPLES::
  
              sage: sZZ3 = (ZZ^3)._sympy_(); sZZ3
              ProductSet(Integers, Integers, Integers)
              sage: (1, 2, 3) in sZZ3
              True
          """
          from sympy import ProductSet
          from sage.interfaces.sympy import sympy_init
          sympy_init()
          return ProductSet(*([self.coordinate_ring()] * self.rank()))
  
  
  ###############################################################################
  #
  # Ambient free modules over an integral domain.
  #
  ###############################################################################
  
  class FreeModule_ambient_domain(FreeModule_ambient):
      """
      Ambient free module over an integral domain.
      """
      def __init__(self, base_ring, rank, sparse=False, coordinate_ring=None):
          """
          Create the ambient free module of given rank over the given
          integral domain.
  
          EXAMPLES::
  
              sage: FreeModule(PolynomialRing(GF(5),'x'), 3)
              Ambient free module of rank 3 over the principal ideal domain
              Univariate Polynomial Ring in x over Finite Field of size 5
          """
          FreeModule_ambient.__init__(self, base_ring,
                  rank, sparse, coordinate_ring)
  
      def _repr_(self):
          """
          The printing representation of self.
  
          EXAMPLES::
  
              sage: R = PolynomialRing(ZZ,'x')
              sage: M = FreeModule(R,7)
              sage: M
              Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring
              sage: print(M._repr_())
              Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring
  
          The system representation can be overwritten, but leaves _repr_
          unmodified.
  
          ::
  
              sage: M.rename('M')
              sage: M
              M
              sage: print(M._repr_())
              Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring
  
          Sparse modules print this fact.
  
          ::
  
              sage: N = FreeModule(R,7,sparse=True)
              sage: N
              Ambient sparse free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring
  
          (Now clean up again.)
  
          ::
  
              sage: M.reset_name()
              sage: M
              Ambient free module of rank 7 over the integral domain Univariate Polynomial Ring in x over Integer Ring
          """
          if self.is_sparse():
              return "Ambient sparse free module of rank %s over the integral domain %s"%(
                  self.rank(), self.base_ring())
          else:
              return "Ambient free module of rank %s over the integral domain %s"%(
                  self.rank(), self.base_ring())
  
      def ambient_vector_space(self):
          """
          Return the ambient vector space, which is this free module
          tensored with its fraction field.
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: V = M.ambient_vector_space(); V
              Vector space of dimension 3 over Rational Field
  
          If an inner product on the module is specified, then this
          is preserved on the ambient vector space.
  
          ::
  
              sage: N = FreeModule(ZZ,4,inner_product_matrix=1)
              sage: U = N.ambient_vector_space()
              sage: U
              Ambient quadratic space of dimension 4 over Rational Field
              Inner product matrix:
              [1 0 0 0]
              [0 1 0 0]
              [0 0 1 0]
              [0 0 0 1]
              sage: P = N.submodule_with_basis([[1,-1,0,0],[0,1,-1,0],[0,0,1,-1]])
              sage: P.gram_matrix()
              [ 2 -1  0]
              [-1  2 -1]
              [ 0 -1  2]
              sage: U == N.ambient_vector_space()
              True
              sage: U == V
              False
          """
          try:
              return self.__ambient_vector_space
          except AttributeError:
              self.__ambient_vector_space = FreeModule(self.base_field(), self.rank(), sparse=self.is_sparse())
              return self.__ambient_vector_space
  
      def coordinate_vector(self, v, check=True):
          """
          Write `v` in terms of the standard basis for ``self`` and
          return the resulting coefficients in a vector over the fraction
          field of the base ring.
  
          INPUT:
  
          - ``v`` -- vector
  
          - ``check`` -- bool (default: True); if True, also verify that
             `v` is really in self.
  
          OUTPUT: list
  
          Returns a vector `c` such that if `B` is the basis for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: V = ZZ^3
              sage: v = V.coordinate_vector([1,5,9]); v
              (1, 5, 9)
              sage: v.parent()
              Vector space of dimension 3 over Rational Field
          """
          # Calling the element constructor directly, since the
          # usual call method indirectly relies on coordinate_vector,
          # hence, an infinite recursion would result
          try:
              out = self.ambient_vector_space()._element_constructor_(v)
          except TypeError:
              raise ArithmeticError("Error transforming the given vector into the ambient vector space")
          if check and out not in self:
              raise ArithmeticError("The given vector does not belong to this free module")
          return out
  
      def vector_space(self, base_field=None):
          """
          Returns the vector space obtained from ``self`` by tensoring with the
          fraction field of the base ring and extending to the field.
  
          EXAMPLES::
  
              sage: M = ZZ^3;  M.vector_space()
              Vector space of dimension 3 over Rational Field
          """
          if base_field is None:
              R = self.base_ring()
              return self.change_ring(R.fraction_field())
          else:
              return self.change_ring(base_field)
  
  ###############################################################################
  #
  # Ambient free modules over a principal ideal domain.
  #
  ###############################################################################
  
  class FreeModule_ambient_pid(FreeModule_generic_pid, FreeModule_ambient_domain):
      """
      Ambient free module over a principal ideal domain.
      """
      def __init__(self, base_ring, rank, sparse=False, coordinate_ring=None):
          """
          Create the ambient free module of given rank over the given
          principal ideal domain.
  
          INPUT:
  
          - ``base_ring`` -- a principal ideal domain
  
          - ``rank`` -- a non-negative integer
  
          - ``sparse`` -- bool (default: False)
  
          - ``coordinate_ring`` -- a ring containing ``base_ring``
            (default: equal to ``base_ring``)
  
          EXAMPLES::
  
              sage: ZZ^3
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
  
          We create the same module with coordinates in ``QQ``::
  
              sage: from sage.modules.free_module import FreeModule_ambient_pid
              sage: M = FreeModule_ambient_pid(ZZ, 3, coordinate_ring=QQ)
              sage: M
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
              sage: v = M.basis()[0]; v
              (1, 0, 0)
              sage: type(v)
              <class 'sage.modules.vector_rational_dense.Vector_rational_dense'>
          """
          FreeModule_ambient_domain.__init__(self, base_ring=base_ring,
                  rank=rank, sparse=sparse, coordinate_ring=coordinate_ring)
  
      def _repr_(self):
          """
          The printing representation of self.
  
          EXAMPLES::
  
              sage: M = FreeModule(ZZ,7)
              sage: M
              Ambient free module of rank 7 over the principal ideal domain Integer Ring
              sage: print(M._repr_())
              Ambient free module of rank 7 over the principal ideal domain Integer Ring
  
          The system representation can be overwritten, but leaves _repr_
          unmodified.
  
          ::
  
              sage: M.rename('M')
              sage: M
              M
              sage: print(M._repr_())
              Ambient free module of rank 7 over the principal ideal domain Integer Ring
  
          Sparse modules print this fact.
  
          ::
  
              sage: N = FreeModule(ZZ,7,sparse=True)
              sage: N
              Ambient sparse free module of rank 7 over the principal ideal domain Integer Ring
  
          (Now clean up again.)
  
          ::
  
              sage: M.reset_name()
              sage: M
              Ambient free module of rank 7 over the principal ideal domain Integer Ring
          """
          if self.is_sparse():
              return "Ambient sparse free module of rank %s over the principal ideal domain %s"%(
                  self.rank(), self.base_ring())
          else:
              return "Ambient free module of rank %s over the principal ideal domain %s"%(
                  self.rank(), self.base_ring())
  
  
  ###############################################################################
  #
  # Ambient free modules over a field (i.e., a vector space).
  #
  ###############################################################################
  
  class FreeModule_ambient_field(FreeModule_generic_field, FreeModule_ambient_pid):
      """
  
      """
      def __init__(self, base_field, dimension, sparse=False):
          """
          Create the ambient vector space of given dimension over the given
          field.
  
          INPUT:
  
  
          -  ``base_field`` - a field
  
          -  ``dimension`` - a non-negative integer
  
          -  ``sparse`` - bool (default: False)
  
  
          EXAMPLES::
  
              sage: QQ^3
              Vector space of dimension 3 over Rational Field
          """
          FreeModule_ambient_pid.__init__(self, base_field, dimension, sparse=sparse)
  
      def _repr_(self):
          """
          The printing representation of self.
  
          EXAMPLES::
  
              sage: V = FreeModule(QQ,7)
              sage: V
              Vector space of dimension 7 over Rational Field
              sage: print(V._repr_())
              Vector space of dimension 7 over Rational Field
  
          The system representation can be overwritten, but leaves _repr_
          unmodified.
  
          ::
  
              sage: V.rename('V')
              sage: V
              V
              sage: print(V._repr_())
              Vector space of dimension 7 over Rational Field
  
          Sparse modules print this fact.
  
          ::
  
              sage: U = FreeModule(QQ,7,sparse=True)
              sage: U
              Sparse vector space of dimension 7 over Rational Field
  
          (Now clean up again.)
  
          ::
  
              sage: V.reset_name()
              sage: V
              Vector space of dimension 7 over Rational Field
          """
          if self.is_sparse():
              return "Sparse vector space of dimension %s over %s"%(self.dimension(), self.base_ring())
          else:
              return "Vector space of dimension %s over %s"%(self.dimension(), self.base_ring())
  
      def ambient_vector_space(self):
          """
          Returns ``self`` as the ambient vector space.
  
          EXAMPLES::
  
              sage: M = QQ^3
              sage: M.ambient_vector_space()
              Vector space of dimension 3 over Rational Field
          """
          return self
  
      def base_field(self):
          """
          Returns the base field of this vector space.
  
          EXAMPLES::
  
              sage: M = QQ^3
              sage: M.base_field()
              Rational Field
          """
          return self.base_ring()
  
      def _element_constructor_(self, e, *args, **kwds):
          """
          Create an element of this vector space.
  
          EXAMPLES::
  
              sage: k.<a> = GF(3^4)
              sage: VS = k.vector_space(map=False)
              sage: VS(a)
              (0, 1, 0, 0)
          """
          try:
              k = e.parent()
              if finite_field.is_FiniteField(k) and k.base_ring() == self.base_ring() and k.degree() == self.degree():
                  return self(e._vector_())
          except AttributeError:
              pass
          return FreeModule_generic_field._element_constructor_(self, e, *args, **kwds)
  
  ###############################################################################
  #
  # R-Submodule of K^n where K is the fraction field of a principal ideal domain $R$.
  #
  ###############################################################################
  
  class FreeModule_submodule_with_basis_pid(FreeModule_generic_pid):
      r"""
      Construct a submodule of a free module over PID with a distinguished basis.
  
      INPUT:
  
      - ``ambient`` -- ambient free module over a principal ideal domain `R`,
        i.e. `R^n`;
  
      - ``basis`` -- list of elements of `K^n`, where `K` is the fraction field
        of `R`. These elements must be linearly independent and will be used as
        the default basis of the constructed submodule;
  
      - ``check`` -- (default: ``True``) if ``False``, correctness of the input
        will not be checked and type conversion may be omitted, use with care;
  
      - ``echelonize`` -- (default:``False``) if ``True``, ``basis`` will be
        echelonized and the result will be used as the default basis of the
        constructed submodule;
  
      - `` echelonized_basis`` -- (default: ``None``) if not ``None``, must be
        the echelonized basis spanning the same submodule as ``basis``;
  
      - ``already_echelonized`` -- (default: ``False``) if ``True``, ``basis``
        must be already given in the echelonized form.
  
      OUTPUT:
  
      - `R`-submodule of `K^n` with the user-specified ``basis``.
  
      EXAMPLES::
  
          sage: M = ZZ^3
          sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W
          Free module of degree 3 and rank 2 over Integer Ring
          User basis matrix:
          [1 2 3]
          [4 5 6]
  
      Now we create a submodule of the ambient vector space, rather than
      ``M`` itself::
  
          sage: W = M.span_of_basis([[1,2,3/2],[4,5,6]]); W
          Free module of degree 3 and rank 2 over Integer Ring
          User basis matrix:
          [  1   2 3/2]
          [  4   5   6]
      """
  
      def __init__(self, ambient, basis, check=True,
          echelonize=False, echelonized_basis=None, already_echelonized=False):
          r"""
          See :class:`FreeModule_submodule_with_basis_pid` for documentation.
  
          TESTS::
  
              sage: M = ZZ^3
              sage: W = M.span_of_basis([[1,2,3],[4,5,6]])
              sage: TestSuite(W).run()
  
          We test that the issue at :trac:`9502` is solved::
  
              sage: parent(W.basis()[0])
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [1 2 3]
              [4 5 6]
              sage: parent(W.echelonized_basis()[0])
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [1 2 3]
              [4 5 6]
  
          Now we test that the issue introduced at :trac:`9502` and reported at
          :trac:`10250` is solved as well::
  
              sage: V = (QQ^2).span_of_basis([[1,1]])
              sage: w = sqrt(2) * V([1,1])
              sage: 3 * w
              (3*sqrt(2), 3*sqrt(2))
          """
          if not isinstance(ambient, FreeModule_ambient_pid):
              raise TypeError("ambient (=%s) must be ambient." % ambient)
          self.__ambient_module = ambient
          R = ambient.base_ring()
          R_coord = R
  
          # Convert all basis elements to the ambient module
          try:
              basis = [ambient(x) for x in basis]
          except TypeError:
              # That failed, try the ambient vector space instead
              V = ambient.ambient_vector_space()
              R_coord = V.base_ring()
              try:
                  basis = [V(x) for x in basis]
              except TypeError:
                  raise TypeError("each element of basis must be in "
                                  "the ambient vector space")
  
          if echelonize and not already_echelonized:
              basis = self._echelonized_basis(ambient, basis)
  
          # The following is WRONG - we should call __init__ of
          # FreeModule_generic_pid. However, it leads to a bunch of errors.
          FreeModule_generic.__init__(self, base_ring=R, coordinate_ring=R_coord,
                                      rank=len(basis), degree=ambient.degree(),
                                      sparse=ambient.is_sparse())
          C = self.element_class
          w = [C(self, x.list(), coerce=False, copy=False) for x in basis]
          self.__basis = basis_seq(self, w)
  
          if echelonize or already_echelonized:
              self.__echelonized_basis = self.__basis
          else:
              if echelonized_basis is None:
                  echelonized_basis = self._echelonized_basis(ambient, basis)
              w = [C(self, x.list(), coerce=False, copy=True)
                   for x in echelonized_basis]
              self.__echelonized_basis = basis_seq(self, w)
          if check and len(basis) != len(self.__echelonized_basis):
              raise ValueError("The given basis vectors must be linearly "
                               "independent.")
  
      def __hash__(self):
          """
          The hash is given by the basis.
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: W = M.span_of_basis([[1,2,3],[4,5,6]])
              sage: hash(W) == hash(W.basis())
              True
          """
          return hash(self.__basis)
  
      def _echelon_matrix_richcmp(self, other, op):
          r"""
          Compare the free module ``self`` with other.
  
          Modules are ordered by their ambient spaces, then by dimension,
          then in order by their echelon matrices.
  
          .. NOTE::
  
             Use :meth:`is_submodule` to determine if one
             module is a submodule of another.
  
          EXAMPLES:
  
          First we compare two equal vector spaces.
  
          ::
  
              sage: from sage.structure.richcmp import op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE
              sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
              sage: W = span([[5,6,7], [8,9,10]], QQ)
              sage: V._echelon_matrix_richcmp(W,op_EQ)
              True
  
          Next we compare a one dimensional space to the two dimensional
          space defined above.
  
          ::
  
              sage: from sage.structure.richcmp import op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE
              sage: M = span([[5,6,7]], QQ)
              sage: V._echelon_matrix_richcmp(M,op_EQ)
              False
              sage: M._echelon_matrix_richcmp(V, op_LT)
              True
              sage: V._echelon_matrix_richcmp(M, op_LT)
              False
  
          We compare a `\ZZ`-module to the one-dimensional
          space above.::
  
              sage: from sage.structure.richcmp import op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE
              sage: V = span([[5,6,7]], ZZ).scale(1/11);  V
              Free module of degree 3 and rank 1 over Integer Ring
              Echelon basis matrix:
              [5/11 6/11 7/11]
              sage: V._echelon_matrix_richcmp(M, op_LT)
              True
              sage: M._echelon_matrix_richcmp(V, op_LT)
              False
          """
          if self is other:
              return rich_to_bool(op, 0)
          if not isinstance(other, FreeModule_generic):
              return NotImplemented
          lx = self.ambient_vector_space()
          rx = other.ambient_vector_space()
          if lx != rx:
              return lx._echelon_matrix_richcmp( rx, op)
  
          lx = self.dimension()
          rx = other.dimension()
          if lx != rx:
              return richcmp_not_equal(lx, rx, op)
  
          lx = self.base_ring()
          rx = other.base_ring()
          if lx != rx:
              return richcmp_not_equal(lx, rx, op)
  
          # We use self.echelonized_basis_matrix() == other.echelonized_basis_matrix()
          # with the matrix to avoid a circular reference.
          return richcmp(self.echelonized_basis_matrix(),
                         other.echelonized_basis_matrix(), op)
  
      def construction(self):
          """
          Returns the functorial construction of self, namely, the subspace
          of the ambient module spanned by the given basis.
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: W = M.span_of_basis([[1,2,3],[4,5,6]]); W
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [1 2 3]
              [4 5 6]
              sage: c, V = W.construction()
              sage: c(V) == W
              True
          """
          from sage.categories.pushout import SubspaceFunctor
          return SubspaceFunctor(self.basis()), self.ambient_module()
  
      def echelonized_basis_matrix(self):
          """
          Return basis matrix for ``self`` in row echelon form.
  
          EXAMPLES::
  
              sage: V = FreeModule(ZZ, 3).span_of_basis([[1,2,3],[4,5,6]])
              sage: V.basis_matrix()
              [1 2 3]
              [4 5 6]
              sage: V.echelonized_basis_matrix()
              [1 2 3]
              [0 3 6]
          """
          try:
              return self.__echelonized_basis_matrix
          except AttributeError:
              pass
          self._echelonized_basis(self.ambient_module(), self.__basis)
          return self.__echelonized_basis_matrix
  
      def _echelonized_basis(self, ambient, basis):
          """
          Given the ambient space and a basis, construct and cache the
          echelonized basis matrix and returns its rows.
  
          N.B. This function is for internal use only!
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: N = M.submodule_with_basis([[1,1,0],[0,2,1]])
              sage: N._echelonized_basis(M,N.basis())
              [(1, 1, 0), (0, 2, 1)]
              sage: V = QQ^3
              sage: W = V.submodule_with_basis([[1,1,0],[0,2,1]])
              sage: W._echelonized_basis(V,W.basis())
              [(1, 0, -1/2), (0, 1, 1/2)]
              sage: V = SR^3
              sage: W = V.submodule_with_basis([[1,0,1]])
              sage: W._echelonized_basis(V,W.basis())
              [(1, 0, 1)]
          """
          # Return the first rank rows (i.e., the nonzero rows).
          d = self._denominator(basis)
          MAT = sage.matrix.matrix_space.MatrixSpace(
              ambient.base_ring(), len(basis), ambient.degree(), sparse = ambient.is_sparse())
          if d != 1:
              basis = [x*d for x in basis]
          A = MAT(basis)
          E = A.echelon_form()
          if d != 1:
              E = E.matrix_over_field()*(~d)   # divide out denominator
          r = E.rank()
          if r < E.nrows():
              E = E.matrix_from_rows(range(r))
          self.__echelonized_basis_matrix = E
          return E.rows()
  
  
  
      def _denominator(self, B):
          """
          The LCM of the denominators of the given list B.
  
          N.B.: This function is for internal use only!
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: L = V.span([[1,1/2,1/3], [-1/5,2/3,3]],ZZ)
              sage: L
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [ 1/5 19/6 37/3]
              [   0 23/6 46/3]
              sage: L._denominator(L.echelonized_basis_matrix().list())
              30
  
          """
          if not B:
              return 1
          d = B[0].denominator()
          from sage.arith.all import lcm
          for x in B[1:]:
              d = lcm(d,x.denominator())
          return d
  
      def _repr_(self):
          """
          The printing representation of self.
  
          EXAMPLES::
  
              sage: L = ZZ^8
              sage: E = L.submodule_with_basis([ L.gen(i) - L.gen(0) for i in range(1,8) ])
              sage: E # indirect doctest
              Free module of degree 8 and rank 7 over Integer Ring
              User basis matrix:
              [-1  1  0  0  0  0  0  0]
              [-1  0  1  0  0  0  0  0]
              [-1  0  0  1  0  0  0  0]
              [-1  0  0  0  1  0  0  0]
              [-1  0  0  0  0  1  0  0]
              [-1  0  0  0  0  0  1  0]
              [-1  0  0  0  0  0  0  1]
  
          ::
  
              sage: M = FreeModule(ZZ,8,sparse=True)
              sage: N = M.submodule_with_basis([ M.gen(i) - M.gen(0) for i in range(1,8) ])
              sage: N # indirect doctest
              Sparse free module of degree 8 and rank 7 over Integer Ring
              User basis matrix:
              [-1  1  0  0  0  0  0  0]
              [-1  0  1  0  0  0  0  0]
              [-1  0  0  1  0  0  0  0]
              [-1  0  0  0  1  0  0  0]
              [-1  0  0  0  0  1  0  0]
              [-1  0  0  0  0  0  1  0]
              [-1  0  0  0  0  0  0  1]
          """
          if self.is_sparse():
              s = "Sparse free module of degree %s and rank %s over %s\n"%(
                  self.degree(), self.rank(), self.base_ring()) + \
                  "User basis matrix:\n%r" % self.basis_matrix()
          else:
              s = "Free module of degree %s and rank %s over %s\n"%(
                  self.degree(), self.rank(), self.base_ring()) + \
                  "User basis matrix:\n%r" % self.basis_matrix()
          return s
  
      def _latex_(self):
          r"""
          Return latex representation of this free module.
  
          EXAMPLES::
  
              sage: A = ZZ^3
              sage: M = A.span_of_basis([[1,2,3],[4,5,6]])
              sage: M._latex_()
              '\\mathrm{RowSpan}_{\\Bold{Z}}\\left(\\begin{array}{rrr}\n1 & 2 & 3 \\\\\n4 & 5 & 6\n\\end{array}\\right)'
          """
          return "\\mathrm{RowSpan}_{%s}%s"%(latex.latex(self.base_ring()), latex.latex(self.basis_matrix()))
  
      def ambient_module(self):
          """
          Return the ambient module related to the `R`-module self,
          which was used when creating this module, and is of the form
          `R^n`. Note that ``self`` need not be contained in the ambient
          module, though ``self`` will be contained in the ambient vector space.
  
          EXAMPLES::
  
              sage: A = ZZ^3
              sage: M = A.span_of_basis([[1,2,'3/7'],[4,5,6]])
              sage: M
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [  1   2 3/7]
              [  4   5   6]
              sage: M.ambient_module()
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
              sage: M.is_submodule(M.ambient_module())
              False
          """
          return self.__ambient_module
  
      def echelon_coordinates(self, v, check=True):
          r"""
          Write `v` in terms of the echelonized basis for self.
  
          INPUT:
  
  
          -  ``v`` - vector
  
          -  ``check`` - bool (default: True); if True, also
             verify that v is really in self.
  
  
          OUTPUT: list
  
          Returns a list `c` such that if `B` is the basis
          for self, then
  
          .. MATH::
  
              \sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: A = ZZ^3
              sage: M = A.span_of_basis([[1,2,'3/7'],[4,5,6]])
              sage: M.coordinates([8,10,12])
              [0, 2]
              sage: M.echelon_coordinates([8,10,12])
              [8, -2]
              sage: B = M.echelonized_basis(); B
              [
              (1, 2, 3/7),
              (0, 3, -30/7)
              ]
              sage: 8*B[0] - 2*B[1]
              (8, 10, 12)
  
          We do an example with a sparse vector space::
  
              sage: V = VectorSpace(QQ,5, sparse=True)
              sage: W = V.subspace_with_basis([[0,1,2,0,0], [0,-1,0,0,-1/2]])
              sage: W.echelonized_basis()
              [
              (0, 1, 0, 0, 1/2),
              (0, 0, 1, 0, -1/4)
              ]
              sage: W.echelon_coordinates([0,0,2,0,-1/2])
              [0, 2]
          """
          if not isinstance(v, free_module_element.FreeModuleElement):
              v = self.ambient_vector_space()(v)
          elif v.degree() != self.degree():
              raise ArithmeticError("vector is not in free module")
          # Find coordinates of v with respect to rref basis.
          E = self.echelonized_basis_matrix()
          P = E.pivots()
          w = v.list_from_positions(P)
          # Next use the transformation matrix from the rref basis
          # to the echelon basis.
          T = self._rref_to_echelon_matrix()
          x = T.linear_combination_of_rows(w).list(copy=False)
          if not check:
              return x
          if v.parent() is self:
              return x
          lc = E.linear_combination_of_rows(x)
          if list(lc) != list(v):
              raise ArithmeticError("vector is not in free module")
          return x
  
      def user_to_echelon_matrix(self):
          """
          Return matrix that transforms a vector written with respect to the
          user basis of ``self`` to one written with respect to the echelon
          basis. The matrix acts from the right, as is usual in Sage.
  
          EXAMPLES::
  
              sage: A = ZZ^3
              sage: M = A.span_of_basis([[1,2,3],[4,5,6]])
              sage: M.echelonized_basis()
              [
              (1, 2, 3),
              (0, 3, 6)
              ]
              sage: M.user_to_echelon_matrix()
              [ 1  0]
              [ 4 -1]
  
          The vector `v=(5,7,9)` in `M` is `(1,1)`
          with respect to the user basis. Multiplying the above matrix on the
          right by this vector yields `(5,-1)`, which has components
          the coordinates of `v` with respect to the echelon basis.
  
          ::
  
              sage: v0,v1 = M.basis(); v = v0+v1
              sage: e0,e1 = M.echelonized_basis()
              sage: v
              (5, 7, 9)
              sage: 5*e0 + (-1)*e1
              (5, 7, 9)
          """
          try:
              return self.__user_to_echelon_matrix
          except AttributeError:
              if self.base_ring().is_field():
                  self.__user_to_echelon_matrix = self._user_to_rref_matrix()
              else:
                  rows = sum([self.echelon_coordinates(b,check=False) for b in self.basis()], [])
                  M = sage.matrix.matrix_space.MatrixSpace(self.base_ring().fraction_field(),
                                               self.dimension(),
                                               sparse = self.is_sparse())
                  self.__user_to_echelon_matrix = M(rows)
          return self.__user_to_echelon_matrix
  
  
      def echelon_to_user_matrix(self):
          """
          Return matrix that transforms the echelon basis to the user basis
          of self. This is a matrix `A` such that if `v` is a
          vector written with respect to the echelon basis for ``self`` then
          `vA` is that vector written with respect to the user basis
          of self.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
              sage: W.echelonized_basis()
              [
              (1, 0, -1),
              (0, 1, 2)
              ]
              sage: A = W.echelon_to_user_matrix(); A
              [-5/3  2/3]
              [ 4/3 -1/3]
  
          The vector `(1,1,1)` has coordinates `v=(1,1)` with
          respect to the echelonized basis for self. Multiplying `vA`
          we find the coordinates of this vector with respect to the user
          basis.
  
          ::
  
              sage: v = vector(QQ, [1,1]); v
              (1, 1)
              sage: v * A
              (-1/3, 1/3)
              sage: u0, u1 = W.basis()
              sage: (-u0 + u1)/3
              (1, 1, 1)
          """
          try:
              return self.__echelon_to_user_matrix
          except AttributeError:
              self.__echelon_to_user_matrix = ~self.user_to_echelon_matrix()
              return self.__echelon_to_user_matrix
  
      def _user_to_rref_matrix(self):
          """
          Returns a transformation matrix from the user specified basis to row
          reduced echelon form, for this module over a PID.
  
          Note: For internal use only! See user_to_echelon_matrix.
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: N = M.submodule_with_basis([[1,1,0],[0,1,1]])
              sage: T = N.user_to_echelon_matrix(); T # indirect doctest
              [1 1]
              [0 1]
              sage: N.basis_matrix()
              [1 1 0]
              [0 1 1]
              sage: N.echelonized_basis_matrix()
              [ 1  0 -1]
              [ 0  1  1]
              sage: T * N.echelonized_basis_matrix() == N.basis_matrix()
              True
          """
          try:
              return self.__user_to_rref_matrix
          except AttributeError:
              A = self.basis_matrix()
              P = self.echelonized_basis_matrix().pivots()
              T = A.matrix_from_columns(P)
              self.__user_to_rref_matrix = T
          return self.__user_to_rref_matrix
  
      def _rref_to_user_matrix(self):
          """
          Returns a transformation matrix from row reduced echelon form to
          the user specified basis, for this module over a PID.
  
          Note: For internal use only! See user_to_echelon_matrix.
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: N = M.submodule_with_basis([[1,1,0],[0,1,1]])
              sage: U = N.echelon_to_user_matrix(); U # indirect doctest
              [ 1 -1]
              [ 0  1]
              sage: N.echelonized_basis_matrix()
              [ 1  0 -1]
              [ 0  1  1]
              sage: N.basis_matrix()
              [1 1 0]
              [0 1 1]
              sage: U * N.basis_matrix() == N.echelonized_basis_matrix()
              True
          """
          try:
              return self.__rref_to_user_matrix
          except AttributeError:
              self.__rref_to_user_matrix = ~self._user_to_rref_matrix()
              return self.__rref_to_user_matrix
  
      def _echelon_to_rref_matrix(self):
          """
          Returns a transformation matrix from the some matrix to the row
          reduced echelon form for this module over a PID.
  
          Note: For internal use only! and not used!
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: N = M.submodule_with_basis([[1,1,0],[1,1,2]])
              sage: N
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [1 1 0]
              [1 1 2]
              sage: T = N._echelon_to_rref_matrix(); T
              [1 0]
              [0 2]
              sage: type(T)
              <class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
              sage: U = N._rref_to_echelon_matrix(); U
              [  1   0]
              [  0 1/2]
              sage: type(U)
              <class 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'>
          """
          try:
              return self.__echelon_to_rref_matrix
          except AttributeError:
              A = self.echelonized_basis_matrix()
              T = A.matrix_from_columns(A.pivots())
              self.__echelon_to_rref_matrix = T
          return self.__echelon_to_rref_matrix
  
      def _rref_to_echelon_matrix(self):
          """
          Returns a transformation matrix from row reduced echelon form to
          some matrix for this module over a PID.
  
          Note: For internal use only!
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: N = M.submodule_with_basis([[1,1,0],[1,1,2]])
              sage: N
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [1 1 0]
              [1 1 2]
              sage: T = N._echelon_to_rref_matrix(); T
              [1 0]
              [0 2]
              sage: type(T)
              <class 'sage.matrix.matrix_integer_dense.Matrix_integer_dense'>
              sage: U = N._rref_to_echelon_matrix(); U
              [  1   0]
              [  0 1/2]
              sage: type(U)
              <class 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'>
          """
          try:
              return self.__rref_to_echelon_matrix
          except AttributeError:
              self.__rref_to_echelon_matrix = ~self._echelon_to_rref_matrix()
              return self.__rref_to_echelon_matrix
  
      def vector_space(self, base_field=None):
          """
          Return the vector space associated to this free module via tensor
          product with the fraction field of the base ring.
  
          EXAMPLES::
  
              sage: A = ZZ^3; A
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
              sage: A.vector_space()
              Vector space of dimension 3 over Rational Field
              sage: M = A.span_of_basis([['1/3',2,'3/7'],[4,5,6]]); M
              Free module of degree 3 and rank 2 over Integer Ring
              User basis matrix:
              [1/3   2 3/7]
              [  4   5   6]
              sage: M.vector_space()
              Vector space of degree 3 and dimension 2 over Rational Field
              User basis matrix:
              [1/3   2 3/7]
              [  4   5   6]
          """
          if base_field is None:
              V = self.ambient_vector_space()
              return V.submodule_with_basis(self.basis())
          return self.change_ring(base_field)
  
      def ambient_vector_space(self):
          """
          Return the ambient vector space in which this free module is
          embedded.
  
          EXAMPLES::
  
              sage: M = ZZ^3;  M.ambient_vector_space()
              Vector space of dimension 3 over Rational Field
  
          ::
  
              sage: N = M.span_of_basis([[1,2,'1/5']])
              sage: N
              Free module of degree 3 and rank 1 over Integer Ring
              User basis matrix:
              [  1   2 1/5]
              sage: M.ambient_vector_space()
              Vector space of dimension 3 over Rational Field
              sage: M.ambient_vector_space() is N.ambient_vector_space()
              True
  
          If an inner product on the module is specified, then this
          is preserved on the ambient vector space.
  
          ::
  
              sage: M = FreeModule(ZZ,4,inner_product_matrix=1)
              sage: V = M.ambient_vector_space()
              sage: V
              Ambient quadratic space of dimension 4 over Rational Field
              Inner product matrix:
              [1 0 0 0]
              [0 1 0 0]
              [0 0 1 0]
              [0 0 0 1]
              sage: N = M.submodule([[1,-1,0,0],[0,1,-1,0],[0,0,1,-1]])
              sage: N.gram_matrix()
              [2 1 1]
              [1 2 1]
              [1 1 2]
              sage: V == N.ambient_vector_space()
              True
          """
          return self.ambient_module().ambient_vector_space()
  
      def basis(self):
          """
          Return the user basis for this free module.
  
          EXAMPLES::
  
              sage: V = ZZ^3
              sage: V.basis()
              [
              (1, 0, 0),
              (0, 1, 0),
              (0, 0, 1)
              ]
              sage: M = V.span_of_basis([['1/8',2,1]])
              sage: M.basis()
              [
              (1/8, 2, 1)
              ]
          """
          return self.__basis
  
      def change_ring(self, R):
          """
          Return the free module over `R` obtained by coercing each
          element of the basis of ``self`` into a vector over the
          fraction field of `R`, then taking the resulting `R`-module.
  
          INPUT:
  
          -  ``R`` - a principal ideal domain
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.subspace([[2, 1/2, 1]])
              sage: W.change_ring(GF(7))
              Vector space of degree 3 and dimension 1 over Finite Field of size 7
              Basis matrix:
              [1 2 4]
  
          ::
  
              sage: M = (ZZ^2) * (1/2)
              sage: N = M.change_ring(QQ)
              sage: N
              Vector space of degree 2 and dimension 2 over Rational Field
              Basis matrix:
              [1 0]
              [0 1]
              sage: N = M.change_ring(QQ['x'])
              sage: N
              Free module of degree 2 and rank 2 over Univariate Polynomial Ring in x over Rational Field
              Echelon basis matrix:
              [1/2   0]
              [  0 1/2]
              sage: N.coordinate_ring()
              Univariate Polynomial Ring in x over Rational Field
  
          The ring must be a principal ideal domain::
  
              sage: M.change_ring(ZZ['x'])
              Traceback (most recent call last):
              ...
              TypeError: the new ring Univariate Polynomial Ring in x over Integer Ring should be a principal ideal domain
          """
          if self.base_ring() is R:
              return self
          if R not in PrincipalIdealDomains():
              raise TypeError("the new ring %r should be a principal ideal domain" % R)
  
          K = R.fraction_field()
          V = VectorSpace(K, self.degree())
          B = [V(b) for b in self.basis()]
          M = self.ambient_module().change_ring(R)
          if self.has_user_basis():
              return M.span_of_basis(B)
          else:
              return M.span(B)
  
      def coordinate_vector(self, v, check=True):
          """
          Write `v` in terms of the user basis for self.
  
          INPUT:
  
          - ``v`` -- vector
  
          - ``check`` -- bool (default: True); if True, also verify that
             `v` is really in self.
  
          OUTPUT: list
  
          Returns a vector `c` such that if `B` is the basis for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: V = ZZ^3
              sage: M = V.span_of_basis([['1/8',2,1]])
              sage: M.coordinate_vector([1,16,8])
              (8)
          """
          # First find the coordinates of v wrt echelon basis.
          w = self.echelon_coordinate_vector(v, check=check)
          # Next use transformation matrix from echelon basis to
          # user basis.
          T = self.echelon_to_user_matrix()
          return T.linear_combination_of_rows(w)
  
      def echelonized_basis(self):
          """
          Return the basis for ``self`` in echelon form.
  
          EXAMPLES::
  
              sage: V = ZZ^3
              sage: M = V.span_of_basis([['1/2',3,1], [0,'1/6',0]])
              sage: M.basis()
              [
              (1/2, 3, 1),
              (0, 1/6, 0)
              ]
              sage: B = M.echelonized_basis(); B
              [
              (1/2, 0, 1),
              (0, 1/6, 0)
              ]
              sage: V.span(B) == M
              True
          """
          return self.__echelonized_basis
  
      def echelon_coordinate_vector(self, v, check=True):
          """
          Write `v` in terms of the echelonized basis for self.
  
          INPUT:
  
  
          -  ``v`` - vector
  
          -  ``check`` - bool (default: True); if True, also
             verify that v is really in self.
  
          Returns a list `c` such that if `B` is the echelonized basis
          for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: V = ZZ^3
              sage: M = V.span_of_basis([['1/2',3,1], [0,'1/6',0]])
              sage: B = M.echelonized_basis(); B
              [
              (1/2, 0, 1),
              (0, 1/6, 0)
              ]
              sage: M.echelon_coordinate_vector(['1/2', 3, 1])
              (1, 18)
          """
          return FreeModule(self.base_ring().fraction_field(), self.rank())(self.echelon_coordinates(v, check=check))
  
      def has_user_basis(self):
          """
          Return ``True`` if the basis of this free module is
          specified by the user, as opposed to being the default echelon
          form.
  
          EXAMPLES::
  
              sage: V = ZZ^3; V.has_user_basis()
              False
              sage: M = V.span_of_basis([[1,3,1]]); M.has_user_basis()
              True
              sage: M = V.span([[1,3,1]]); M.has_user_basis()
              False
          """
          return True
  
      def linear_combination_of_basis(self, v):
          """
          Return the linear combination of the basis for ``self`` obtained from
          the coordinates of v.
  
          INPUT:
  
          - ``v`` - list
  
          EXAMPLES::
  
              sage: V = span([[1,2,3], [4,5,6]], ZZ); V
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [1 2 3]
              [0 3 6]
              sage: V.linear_combination_of_basis([1,1])
              (1, 5, 9)
  
          This should raise an error if the resulting element is not in self::
  
              sage: W = (QQ**2).span([[2, 0], [0, 8]], ZZ)
              sage: W.linear_combination_of_basis([1, -1/2])
              Traceback (most recent call last):
              ...
              TypeError: element [2, -4] is not in free module
          """
          R = self.base_ring()
          check = (not R.is_field()) and any(a not in R for a in list(v))
          return self(self.basis_matrix().linear_combination_of_rows(v),
                      check=check, copy=False, coerce=False)
  
  
  class FreeModule_submodule_pid(FreeModule_submodule_with_basis_pid):
      """
      An `R`-submodule of `K^n` where `K` is the
      fraction field of a principal ideal domain `R`.
  
      EXAMPLES::
  
          sage: M = ZZ^3
          sage: W = M.span_of_basis([[1,2,3],[4,5,19]]); W
          Free module of degree 3 and rank 2 over Integer Ring
          User basis matrix:
          [ 1  2  3]
          [ 4  5 19]
  
      Generic tests, including saving and loading submodules and elements::
  
          sage: TestSuite(W).run()
          sage: v = W.0 + W.1
          sage: TestSuite(v).run()
      """
      def __init__(self, ambient, gens, check=True, already_echelonized=False):
          """
          Create an embedded free module over a PID.
  
          EXAMPLES::
  
              sage: V = ZZ^3
              sage: W = V.span([[1,2,3],[4,5,6]])
              sage: W
              Free module of degree 3 and rank 2 over Integer Ring
              Echelon basis matrix:
              [1 2 3]
              [0 3 6]
          """
          FreeModule_submodule_with_basis_pid.__init__(self, ambient, basis=gens,
              echelonize=True, already_echelonized=already_echelonized)
  
      def _repr_(self):
          """
          The printing representation of self.
  
          EXAMPLES::
  
              sage: M = ZZ^8
              sage: L = M.submodule([ M.gen(i) - M.gen(0) for i in range(1,8) ])
              sage: L # indirect doctest
              Free module of degree 8 and rank 7 over Integer Ring
              Echelon basis matrix:
              [ 1  0  0  0  0  0  0 -1]
              [ 0  1  0  0  0  0  0 -1]
              [ 0  0  1  0  0  0  0 -1]
              [ 0  0  0  1  0  0  0 -1]
              [ 0  0  0  0  1  0  0 -1]
              [ 0  0  0  0  0  1  0 -1]
              [ 0  0  0  0  0  0  1 -1]
          """
          if self.is_sparse():
              s = "Sparse free module of degree %s and rank %s over %s\n"%(
                  self.degree(), self.rank(), self.base_ring()) + \
                  "Echelon basis matrix:\n%s"%self.basis_matrix()
          else:
              s = "Free module of degree %s and rank %s over %s\n"%(
                  self.degree(), self.rank(), self.base_ring()) + \
                  "Echelon basis matrix:\n%s"%self.basis_matrix()
          return s
  
      def coordinate_vector(self, v, check=True):
          """
          Write `v` in terms of the user basis for self.
  
          INPUT:
  
          - ``v`` -- vector
  
          - ``check`` -- bool (default: True); if True, also verify that
             `v` is really in self.
  
          OUTPUT: list
  
          Returns a list `c` such that if `B` is the basis for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: V = ZZ^3
              sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
              sage: W.coordinate_vector([1,5,9])
              (5, -1)
          """
          return self.echelon_coordinate_vector(v, check=check)
  
      def has_user_basis(self):
          r"""
          Return ``True`` if the basis of this free module is
          specified by the user, as opposed to being the default echelon
          form.
  
          EXAMPLES::
  
              sage: A = ZZ^3; A
              Ambient free module of rank 3 over the principal ideal domain Integer Ring
              sage: A.has_user_basis()
              False
              sage: W = A.span_of_basis([[2,'1/2',1]])
              sage: W.has_user_basis()
              True
              sage: W = A.span([[2,'1/2',1]])
              sage: W.has_user_basis()
              False
          """
          return False
  
  
  class FreeModule_submodule_with_basis_field(FreeModule_generic_field, FreeModule_submodule_with_basis_pid):
      """
      An embedded vector subspace with a distinguished user basis.
  
      EXAMPLES::
  
          sage: M = QQ^3; W = M.submodule_with_basis([[1,2,3], [4,5,19]]); W
          Vector space of degree 3 and dimension 2 over Rational Field
          User basis matrix:
          [ 1  2  3]
          [ 4  5 19]
  
      Since this is an embedded vector subspace with a distinguished user
      basis possibly different than the echelonized basis, the
      echelon_coordinates() and user coordinates() do not agree::
  
          sage: V = QQ^3
  
      ::
  
          sage: W = V.submodule_with_basis([[1,2,3], [4,5,6]])
          sage: W
          Vector space of degree 3 and dimension 2 over Rational Field
          User basis matrix:
          [1 2 3]
          [4 5 6]
  
      ::
  
          sage: v = V([1,5,9])
          sage: W.echelon_coordinates(v)
          [1, 5]
          sage: vector(QQ, W.echelon_coordinates(v)) * W.echelonized_basis_matrix()
          (1, 5, 9)
  
      ::
  
          sage: v = V([1,5,9])
          sage: W.coordinates(v)
          [5, -1]
          sage: vector(QQ, W.coordinates(v)) * W.basis_matrix()
          (1, 5, 9)
  
      Generic tests, including saving and loading submodules and elements::
  
          sage: TestSuite(W).run()
  
          sage: K.<x> = FractionField(PolynomialRing(QQ,'x'))
          sage: M = K^3; W = M.span_of_basis([[1,1,x]])
          sage: TestSuite(W).run()
      """
      def __init__(self, ambient, basis, check=True,
          echelonize=False, echelonized_basis=None, already_echelonized=False):
          """
          Create a vector space with given basis.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
              sage: W
              Vector space of degree 3 and dimension 2 over Rational Field
              User basis matrix:
              [1 2 3]
              [4 5 6]
          """
          FreeModule_submodule_with_basis_pid.__init__(
              self, ambient, basis=basis, check=check, echelonize=echelonize,
              echelonized_basis=echelonized_basis, already_echelonized=already_echelonized)
  
      def _repr_(self):
          """
          The printing representation of self.
  
          EXAMPLES::
  
              sage: V = VectorSpace(QQ,5)
              sage: U = V.submodule([ V.gen(i) - V.gen(0) for i in range(1,5) ])
              sage: U # indirect doctest
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
              sage: print(U._repr_())
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
  
          The system representation can be overwritten, but leaves _repr_
          unmodified.
  
          ::
  
              sage: U.rename('U')
              sage: U
              U
              sage: print(U._repr_())
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
  
          Sparse vector spaces print this fact.
  
          ::
  
              sage: VV = VectorSpace(QQ,5,sparse=True)
              sage: UU = VV.submodule([ VV.gen(i) - VV.gen(0) for i in range(1,5) ])
              sage: UU # indirect doctest
              Sparse vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
  
          (Now clean up again.)
  
          ::
  
              sage: U.reset_name()
              sage: U
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
          """
          if self.is_sparse():
              return "Sparse vector space of degree %s and dimension %s over %s\n"%(
                      self.degree(), self.dimension(), self.base_field()) + \
                      "User basis matrix:\n%r" % self.basis_matrix()
          else:
              return "Vector space of degree %s and dimension %s over %s\n"%(
                      self.degree(), self.dimension(), self.base_field()) + \
                      "User basis matrix:\n%r" % self.basis_matrix()
  
      def _denominator(self, B):
          """
          Given a list (of field elements) returns 1 as the common
          denominator.
  
          N.B.: This function is for internal use only!
  
          EXAMPLES::
  
              sage: U = QQ^3
              sage: U
              Vector space of dimension 3 over Rational Field
              sage: U.denominator()
              1
              sage: V = U.span([[1,1/2,1/3], [-1/5,2/3,3]])
              sage: V
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [   1    0 -5/3]
              [   0    1    4]
              sage: W = U.submodule_with_basis([[1,1/2,1/3], [-1/5,2/3,3]])
              sage: W
              Vector space of degree 3 and dimension 2 over Rational Field
              User basis matrix:
              [   1  1/2  1/3]
              [-1/5  2/3    3]
              sage: W._denominator(W.echelonized_basis_matrix().list())
              1
          """
          return 1
  
      def _echelonized_basis(self, ambient, basis):
          """
          Given the ambient space and a basis, construct and cache the
          echelonized basis matrix and returns its rows.
  
          N.B. This function is for internal use only!
  
          EXAMPLES::
  
              sage: M = ZZ^3
              sage: N = M.submodule_with_basis([[1,1,0],[0,2,1]])
              sage: N._echelonized_basis(M,N.basis())
              [(1, 1, 0), (0, 2, 1)]
              sage: V = QQ^3
              sage: W = V.submodule_with_basis([[1,1,0],[0,2,1]])
              sage: W._echelonized_basis(V,W.basis())
              [(1, 0, -1/2), (0, 1, 1/2)]
          """
          MAT = sage.matrix.matrix_space.MatrixSpace(
              base_ring=ambient.base_ring(),
              nrows=len(basis), ncols=ambient.degree(),
              sparse=ambient.is_sparse())
          A = MAT(basis)
          E = A.echelon_form()
          # Return the first rank rows (i.e., the nonzero rows).
          return E.rows()[:E.rank()]
  
      def is_ambient(self):
          """
          Return False since this is not an ambient module.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: V.is_ambient()
              True
              sage: W = V.span_of_basis([[1,2,3],[4,5,6]])
              sage: W.is_ambient()
              False
          """
          return False
  
  
  class FreeModule_submodule_field(FreeModule_submodule_with_basis_field):
      """
      An embedded vector subspace with echelonized basis.
  
      EXAMPLES:
  
      Since this is an embedded vector subspace with echelonized basis,
      the echelon_coordinates() and user coordinates() agree::
  
          sage: V = QQ^3
          sage: W = V.span([[1,2,3],[4,5,6]])
          sage: W
          Vector space of degree 3 and dimension 2 over Rational Field
          Basis matrix:
          [ 1  0 -1]
          [ 0  1  2]
  
      ::
  
          sage: v = V([1,5,9])
          sage: W.echelon_coordinates(v)
          [1, 5]
          sage: vector(QQ, W.echelon_coordinates(v)) * W.basis_matrix()
          (1, 5, 9)
          sage: v = V([1,5,9])
          sage: W.coordinates(v)
          [1, 5]
          sage: vector(QQ, W.coordinates(v)) * W.basis_matrix()
          (1, 5, 9)
      """
      def __init__(self, ambient, gens, check=True, already_echelonized=False):
          """
          Create an embedded vector subspace with echelonized basis.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.span([[1,2,3],[4,5,6]])
              sage: W
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
          """
          if is_FreeModule(gens):
              gens = gens.gens()
          FreeModule_submodule_with_basis_field.__init__(self, ambient, basis=gens, check=check,
              echelonize=not already_echelonized, already_echelonized=already_echelonized)
  
      def _repr_(self):
          """
          The default printing representation of self.
  
          EXAMPLES::
  
              sage: V = VectorSpace(QQ,5)
              sage: U = V.submodule([ V.gen(i) - V.gen(0) for i in range(1,5) ])
              sage: U # indirect doctest
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
              sage: print(U._repr_())
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
  
          The system representation can be overwritten, but leaves _repr_
          unmodified.
  
          ::
  
              sage: U.rename('U')
              sage: U
              U
              sage: print(U._repr_())
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
  
          Sparse vector spaces print this fact.
  
          ::
  
              sage: VV = VectorSpace(QQ,5,sparse=True)
              sage: UU = VV.submodule([ VV.gen(i) - VV.gen(0) for i in range(1,5) ])
              sage: UU # indirect doctest
              Sparse vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
  
          (Now clean up again.)
  
          ::
  
              sage: U.reset_name()
              sage: U
              Vector space of degree 5 and dimension 4 over Rational Field
              Basis matrix:
              [ 1  0  0  0 -1]
              [ 0  1  0  0 -1]
              [ 0  0  1  0 -1]
              [ 0  0  0  1 -1]
          """
          if self.is_sparse():
              return "Sparse vector space of degree %s and dimension %s over %s\n"%(
                  self.degree(), self.dimension(), self.base_field()) + \
                  "Basis matrix:\n%r" % self.basis_matrix()
          else:
              return "Vector space of degree %s and dimension %s over %s\n"%(
                  self.degree(), self.dimension(), self.base_field()) + \
                  "Basis matrix:\n%r" % self.basis_matrix()
  
      def echelon_coordinates(self, v, check=True):
          """
          Write `v` in terms of the echelonized basis of self.
  
          INPUT:
  
  
          -  ``v`` - vector
  
          -  ``check`` - bool (default: True); if True, also
             verify that v is really in self.
  
  
          OUTPUT: list
  
          Returns a list `c` such that if `B` is the basis for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.span([[1,2,3],[4,5,6]])
              sage: W
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
  
          ::
  
              sage: v = V([1,5,9])
              sage: W.echelon_coordinates(v)
              [1, 5]
              sage: vector(QQ, W.echelon_coordinates(v)) * W.basis_matrix()
              (1, 5, 9)
          """
          if not isinstance(v, free_module_element.FreeModuleElement):
              v = self.ambient_vector_space()(v)
          if v.degree() != self.degree():
              raise ArithmeticError("v (=%s) is not in self"%v)
          E = self.echelonized_basis_matrix()
          P = E.pivots()
          if not P:
              if check and v != 0:
                  raise ArithmeticError("vector is not in free module")
              return []
          w = v.list_from_positions(P)
          if not check:
              # It's really really easy.
              return w
          if v.parent() is self:   # obvious that v is really in here.
              return w
          # the "linear_combination_of_rows" call dominates the runtime
          # of this function, in the check==False case when the parent
          # of v is not self.
          lc = E.linear_combination_of_rows(w)
          if lc.list() != v.list():
              raise ArithmeticError("vector is not in free module")
          return w
  
      def coordinate_vector(self, v, check=True):
          """
          Write `v` in terms of the user basis for self.
  
          INPUT:
  
          - ``v`` -- vector
  
          - ``check`` -- bool (default: True); if True, also verify that
             `v` is really in self.
  
          OUTPUT: list
  
          Returns a list `c` such that if `B` is the basis for self, then
  
          .. MATH::
  
              \\sum c_i B_i = v.
  
          If `v` is not in self, raise an ``ArithmeticError`` exception.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.span([[1,2,3],[4,5,6]]); W
              Vector space of degree 3 and dimension 2 over Rational Field
              Basis matrix:
              [ 1  0 -1]
              [ 0  1  2]
              sage: v = V([1,5,9])
              sage: W.coordinate_vector(v)
              (1, 5)
              sage: W.coordinates(v)
              [1, 5]
              sage: vector(QQ, W.coordinates(v)) * W.basis_matrix()
              (1, 5, 9)
  
          ::
  
              sage: V = VectorSpace(QQ,5, sparse=True)
              sage: W = V.subspace([[0,1,2,0,0], [0,-1,0,0,-1/2]])
              sage: W.coordinate_vector([0,0,2,0,-1/2])
              (0, 2)
          """
          return self.echelon_coordinate_vector(v, check=check)
  
      def has_user_basis(self):
          """
          Return ``True`` if the basis of this free module is
          specified by the user, as opposed to being the default echelon
          form.
  
          EXAMPLES::
  
              sage: V = QQ^3
              sage: W = V.subspace([[2,'1/2', 1]])
              sage: W.has_user_basis()
              False
              sage: W = V.subspace_with_basis([[2,'1/2',1]])
              sage: W.has_user_basis()
              True
          """
          return False
  
  def basis_seq(V, vecs):
      """
      This converts a list vecs of vectors in V to an Sequence of
      immutable vectors.
  
      Should it? I.e. in most ``other`` parts of the system the return type
      of basis or generators is a tuple.
  
      EXAMPLES::
  
          sage: V = VectorSpace(QQ,2)
          sage: B = V.gens()
          sage: B
          ((1, 0), (0, 1))
          sage: v = B[0]
          sage: v[0] = 0 # immutable
          Traceback (most recent call last):
          ...
          ValueError: vector is immutable; please change a copy instead (use copy())
          sage: sage.modules.free_module.basis_seq(V, V.gens())
          [
          (1, 0),
          (0, 1)
          ]
      """
      for z in vecs:
          z.set_immutable()
      return Sequence(vecs, universe=V, check = False, immutable=True, cr=True)
  
  
  class RealDoubleVectorSpace_class(FreeModule_ambient_field):
      def __init__(self,n):
          FreeModule_ambient_field.__init__(self,sage.rings.real_double.RDF,n)
  
      def coordinates(self,v):
          return v
  
  class ComplexDoubleVectorSpace_class(FreeModule_ambient_field):
      def __init__(self,n):
          FreeModule_ambient_field.__init__(self,sage.rings.complex_double.CDF,n)
  
      def coordinates(self,v):
          return v
  
  
  
  ######################################################
  
  def element_class(R, is_sparse):
      """
      The class of the vectors (elements of a free module) with base ring
      R and boolean is_sparse.
  
      EXAMPLES::
  
          sage: FF = FiniteField(2)
          sage: P = PolynomialRing(FF,'x')
          sage: sage.modules.free_module.element_class(QQ, is_sparse=True)
          <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
          sage: sage.modules.free_module.element_class(QQ, is_sparse=False)
          <class 'sage.modules.vector_rational_dense.Vector_rational_dense'>
          sage: sage.modules.free_module.element_class(ZZ, is_sparse=True)
          <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
          sage: sage.modules.free_module.element_class(ZZ, is_sparse=False)
          <class 'sage.modules.vector_integer_dense.Vector_integer_dense'>
          sage: sage.modules.free_module.element_class(FF, is_sparse=True)
          <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
          sage: sage.modules.free_module.element_class(FF, is_sparse=False)
          <class 'sage.modules.vector_mod2_dense.Vector_mod2_dense'>
          sage: sage.modules.free_module.element_class(GF(7), is_sparse=False)
          <class 'sage.modules.vector_modn_dense.Vector_modn_dense'>
          sage: sage.modules.free_module.element_class(P, is_sparse=True)
          <class 'sage.modules.free_module_element.FreeModuleElement_generic_sparse'>
          sage: sage.modules.free_module.element_class(P, is_sparse=False)
          <class 'sage.modules.free_module_element.FreeModuleElement_generic_dense'>
      """
      import sage.modules.vector_real_double_dense
      import sage.modules.vector_complex_double_dense
  
      if sage.rings.integer_ring.is_IntegerRing(R) and not is_sparse:
          from .vector_integer_dense import Vector_integer_dense
          return Vector_integer_dense
      elif sage.rings.rational_field.is_RationalField(R) and not is_sparse:
          from .vector_rational_dense import Vector_rational_dense
          return Vector_rational_dense
      elif sage.rings.finite_rings.integer_mod_ring.is_IntegerModRing(R) and not is_sparse:
          from .vector_mod2_dense import Vector_mod2_dense
          if R.order() == 2:
              return Vector_mod2_dense
          from .vector_modn_dense import Vector_modn_dense, MAX_MODULUS
          if R.order() < MAX_MODULUS:
              return Vector_modn_dense
          else:
              return free_module_element.FreeModuleElement_generic_dense
      elif sage.rings.real_double.is_RealDoubleField(R) and not is_sparse:
          return sage.modules.vector_real_double_dense.Vector_real_double_dense
      elif sage.rings.complex_double.is_ComplexDoubleField(R) and not is_sparse:
          return sage.modules.vector_complex_double_dense.Vector_complex_double_dense
      elif sage.symbolic.ring.is_SymbolicExpressionRing(R) and not is_sparse:
          import sage.modules.vector_symbolic_dense
          return sage.modules.vector_symbolic_dense.Vector_symbolic_dense
      elif sage.symbolic.callable.is_CallableSymbolicExpressionRing(R) and not is_sparse:
          import sage.modules.vector_callable_symbolic_dense
          return sage.modules.vector_callable_symbolic_dense.Vector_callable_symbolic_dense
      else:
          if is_sparse:
              return free_module_element.FreeModuleElement_generic_sparse
          else:
              return free_module_element.FreeModuleElement_generic_dense
      raise NotImplementedError
  
  @richcmp_method
  class EchelonMatrixKey(object):
      r"""
      A total ordering on free modules for sorting.
  
      This class orders modules by their ambient spaces, then by dimension,
      then in order by their echelon matrices. If a function returns a list
      of free modules, this can be used to sort the output and thus render
      it deterministic.
  
      INPUT:
  
      - ``obj`` -- a free module
  
      EXAMPLES::
  
          sage: V = span([[1,2,3], [5,6,7], [8,9,10]], QQ)
          sage: W = span([[5,6,7], [8,9,10]], QQ)
          sage: X = span([[5,6,7]], ZZ).scale(1/11)
          sage: Y = CC^3
          sage: Z = ZZ^2
          sage: modules = [V,W,X,Y,Z]
          sage: modules_sorted = [Z,X,V,W,Y]
          sage: from sage.modules.free_module import EchelonMatrixKey
          sage: modules.sort(key=EchelonMatrixKey)
          sage: modules == modules_sorted
          True
      """
      def __init__(self, obj):
          r"""
          Create a container for a free module with a total ordering.
  
          EXAMPLES::
  
              sage: R.<x> = QQ[]
              sage: V = span(R,[[x,1+x],[x^2,2+x]])
              sage: W = R^2
              sage: from sage.modules.free_module import EchelonMatrixKey
              sage: V = EchelonMatrixKey(V)
              sage: W = EchelonMatrixKey(W)
              sage: V < W
              False
          """
          self.obj = obj
  
      def __richcmp__(self, other, op):
          r"""
          A total ordering on free modules.
  
          TESTS::
  
              sage: from sage.modules.free_module import EchelonMatrixKey
              sage: Y = EchelonMatrixKey(CC^3)
              sage: Z = EchelonMatrixKey(ZZ^2)
              sage: Z < Y
              True
          """
          return self.obj._echelon_matrix_richcmp(other.obj, op)
  
  