# When generating set is not a basis

If a generating set of a vector space being made up of linearly independent vectors constitues a basis, when such a set is not a basis does it mean that its vectors are linearly dependent?

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Yes. Let $S$ be the generating set and let $B\subset S$ be a basis. If $B,S$ are not equal, then there exists $u\in S-B$ since $B$ is a basis, it follows that $u=c_1v_1+c_2v_2+...+c_nv_n$ for some $v_1,v_2,...,v_n\in B$. From this it follows that {$u,v_1,...,v_n$} are linearly dependent. Hence, $S$ is not linearly independent.