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 \({latex.inline[v_{1}, ..., v_{n}](v_{1}, ..., v_{n})} spans V. The list \){latex.inlinev{1}, ..., v{n}} can be reduced to a basis of V per 1753318207 - Axler 2.30 Every spanning list contains a basis|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.