(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. If V is spanned by {v1,v2, ..., vk} and one of these vectors can be written as a linear combination of the other k-1 vectors, prove that the span of these k-1 vectors is also V.

2. Relevant equations

A set S = {v1,v2, ..., vk}, k >= 2 is linearly dependent if and only if at least one of the vectors vj can be written as a linear combination of the other vectors in S.

3. The attempt at a solution

Since one of the vectors can be written as a linear combination of the other k-1 vectors, this means that the set of vectors is linearly dependent. Also, since V is spanned by {v1,v2, ..., vk}, then span(S) = {c1v1 + c2v2 + ...+ ckvk : c1, c2, ..., ck are real numbers} by the definition of the span of a set. In addition, by the definition of linearly dependent, there exists a nontrivial solution to the c1v1 + c2v2 +...+ ckvk = 0. These are all the pieces of information that I have deduced from the information given in the problem. From here, I am unsure as to how to proceed in this proof.

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# Homework Help: Linear algebra proof span

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