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 $(V,+,\cdot )$, and two Hamel Basis $\{b_i{\}}_{i=1}^m,\{b_i^{\prime }{\}}_{j=1}^n\subset V$, we have that they’re both Linearly Independent and both Spanning Sets.

Thus we have that $m\le n$ and $n\le m$. Thus, they both have the same number of elements.

Properties

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