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.

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