the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below. Basic constructions In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal properties, which determine an object by specifying the linear maps from to any other vector space. Subspaces and quotient spaces A nonempty subset of a vector space that is closed under addition and scalar multiplication (and therefore contains the -vector of ) is called a linear subspace of or simply a subspace of when the ambient space is unambiguously a vector space. Subspaces of are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called its span, and it is the smallest subspace of containing the set Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension a vector hyperplane is thus a subspace of dimension The counterpart to subspaces are quotient vector spaces. Given any subspace the quotient space (" modulo ") is defined as follows: as a set, it consists of where is an arbitrary vector in The sum of two such elements and is and scalar multiplication is given by The key point in this definition is that if and only if the difference of and lies in This way, the quotient space "forgets" information that is contained in the subspace The kernel of a linear map consists of vectors that are mapped to in The kernel and the image are subspaces of and respectively. The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field ) is an abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups. Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms) and