Theorem

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

Proof

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