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.inlinev_{1}, ..., v_{n}} is linearly independent in V. The list ${latex.inlinev_{1}, ..., v_{n}} can be extended to a basis of V by 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.