Theorem

Any two bases of a finite dimensional vector space have the same length.

Proof

Suppose V is finite dimensional. Let ${latex.inlineB_{1}} and ${latex.inlineB_{2}} be two bases of V. Then ${latex.inlineB_{1}} is linearly independent and ${latex.inlineB_{2}} spans V, so the length of ${latex.inlineB_{1}} is at most the length of ${latex.inlineB_{2}} per [2.22](<1753232272 - Axler 2.22 Length of linearly independent list <= length of spanning list>). Interchange the roles of ${latex.inlineB_{1}} and ${latex.inlineB_{2}}. And you get the same thing. Thus the length of the two bases are equal as desired.