Proposition (Dependence Lemma)

Let $V$ be a vector space and suppose that $v_1,..,v_k\in V$ are linearly dependent. There is a $j\in\{1,...,k\}$ such that (a) $v_j\in span(v_1,...,v_{j-1})$, and (b) $span(v_1,...,v_{j-1},v_{j+1},...,v_k)=span(v_1,...,v_k)$