Chapter 4.5, Problem 82E

Proof Prove Theorem 4.12.

THEOREM 4.12 Basis Tests in an $\mathit{n}$-Dimensional Space

Let $V$ be a vector space of dimension $n$.

1. If $S=\left\{v_1,v_2,\dots ,v_n\right\}$ is a linearly independent set of vectors in $V$, then $S$ is a basis for $V$.

2. If $S=\left\{v_1,v_2,\dots ,v_n\right\}$ spans $V$, then $S$ is a basis for $V$.