Dimension theorem for vector spaces

The dimension theorem for vector spaces is the following:

Given a vector space V, any two linearly independent generating sets (in other words, any two bases) have the same cardinality.

If V is finitely generated, the result says that any two bases have the same number of elements.

The cardinality of a basis is called the dimension of the vector space.

The proof in the general case makes use of Zorn's lemma (or equivalently, the axiom of choice), while for the finitely generated case it can be done with elementary arguments of linear algebra.

See also: Basis (linear algebra)

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