Theorem

Suppose V is finite dimensional. Then every spanning 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}} spans V. The list ${latex.inlinev_{1}, ..., v_{n}} can be reduced to a basis of V per 2.30. However, every basis of V has length n, so in this case the reduction is the trivial one, meaning no elements are deleted. Thus, the list of v’s is a basis of V as desired.