All Hamel bases have the same number of elements (in a finitely generated vector space)

up:: Steinitz Exchange Lemma

Given a Finitely Generated Vector Space , and two Hamel Basis , we have that they’re both Linearly Independent and both Spanning Sets.

Thus we have that and . Thus, they both have the same number of elements.

Properties

One can thus define the Finitely Generated Vector Space Dimension as the cardinality of its bases, since it is unambiguous.