## A fact about linear independence

Let $F$ be a field and let $V$ be an $F$-vector space. Suppose $B = \{b_i\} \subseteq V$. Prove that $B$ is linearly dependent over $F$ if and only if some $b_i$ is an $F$-linear combination of the others.

Suppose $B$ is linearly dependent; then we have $\alpha_i \in F$ such that $\sum \alpha_i b_i = 0$, with some $\alpha_i$ nonzero. Suppose without loss of generality that $\alpha_1 \neq 0$. Now $-\alpha_1b_1 = \sum_{i \neq 1} \alpha_ib_i$, so that $b_1 = \sum_{i \neq 1} \frac{-\alpha_i}{\alpha_1} b_i$.

Conversely, suppose (without loss of generality) that $b_1 = \sum_{i \neq 1} \alpha_i b_i$. If the $\alpha_i$ are all zero, then $b_1 = 0$, and certainly $B$ is linearly dependent. If some $\alpha_i$ is nonzero, then $0 = -b_1 + \sum_{i \neq 1} \alpha_i b_i$, and so $B$ is linearly dependent.