Prove the following theorem, which could be viewed as a reformulation of parts (3) and (4) of Theorem G, or more appropriately as a corollary of Theorem G (Proof Technique LC).

Suppose $V$ is a vector space and $S$ is a subset of $V$ such that the number of vectors in $S$ equals the dimension of $V$. Then $S$ is linearly independent if and only if $S$ spans $V$.