for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied.The closure property also implies that every intersection of linear subspaces is a linear subspace. Linear span Given a subset of a vector space , the linear span or simply the span of is the smallest linear subspace of that contains , in the sense that it is the intersection of all linear subspaces that contain . The span of is also the set of all linear combinations of elements of . If is the span of , one says that spans or generates , and that is a spanning set or a generating set of . Basis and dimension A subset of a vector space is a basis if its elements are linearly independent and span the vector space. Every vector space has at least one basis, generally many (see ). Moreover, all bases of a vector space have the same cardinality, which is called the dimension of the vector space (see Dimension theorem for vector spaces). This is a fundamental property of vector spaces, which is detailed in the remainder of the section. Bases are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called Hamel bases, depend on the axiom of choice. It follows that, in general, no base can be explicitly described. For example, the real numbers form an infinite-dimensional vector space over the rational numbers, for which no specific basis is known. Consider a basis of a vector space of dimension over a field . The definition of a basis implies that every may be written with in , and that this decomposition is unique. The scalars are called the coordinates of on the basis. They are also said to be the coefficients of the decomposition of on the basis. One also says that the -tuple of the coordinates is the coordinate vector of on the basis, since the set of the -tuples of elements of is a vector space for componentwise addition and scalar multiplication, whose dimension is . The one-to-one correspondence between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a vector space isomorphism, which allows translating reasonings and computations on vectors into