Theorem

Suppose V is finite dimensional. Then every linearly independent list of vectors in V of length dim V is a basis of V.

Proof

Suppose dim V = n and \({latex.inline[v_{1}, ..., v_{n}](v_{1}, ..., v_{n})} is linearly independent in V. The list \){latex.inlinev{1}, ..., v{n}} can be extended to a basis of V by 1753318250 - Axler 2.32 Every linearly independent list extends to a basis|2.32. However, every basis of V has the length n, so the extension is the trivial one, meaning no elements are added. Thus the list of v’s is a basis as desird.