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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.

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A basis is a generating set or spanning set that is linearly independent. If a set spans the vector space but is not a basis, then it must be linearly dependent.

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Great! So that's true, could you also tell me what kind of logical reasoning is this? – Matteo Dec 4 '12 at 23:13
This proves me how much better would be adopting the middle ages approach to teaching, when logical reasoning ability was considered the ground to which every other science had to lay its foundations (and also as Kurt Godel stated) – Matteo Dec 4 '12 at 23:18

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