1. The problem statement, all variables and given/known data Prove the following theorem: Let (v1, . . . , vk) be a sequence of vectors from a vector space V . Prove that the sequence if linearly dependent if and only if for some j, 1 ≤ j ≤ k, vj is a linear combination of (v1, . . . , vk) − (vj ). 2. Relevant equations 3. The attempt at a solution the if and only if is what bothers me. I know how to prove the following direction: If vj is a linear combination of (v1,.....,vk)-vj then linearly dependent My approach is if c1v1+.....+ckvk=vj then c1v1+....+ckvk-vj=0, so there exists a set of constants c1,..,ck,cj=-1 such that c1v1+...+ckvk=0 (note c1,...,ck can't all be zero) is that right? I don't know how to show if linearly dependent then vj is a linear combination of (v1,.....,vk)-vj, I guess the contrapositive would be ok if for all vj, vj is not a linear combination of (v1,...,vk)-vj then the sequence of vectors is not linearly independent. Proof by contradiction Assume to the contrary that there exists a vj such that vj is a linear combination of (v1,..,vk)-vj and the sequence of vectors is not linearly independent. then there exists a set of constants such that c1v1+.....+ckvk=vj ,so c1v1+.....+ckvk-vj=0 which shows the system is linearly dependent, so contradiction ..