- #1

Danielm

- 21

- 0

## Homework Statement

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 ).

## Homework Equations

## 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

..

Last edited: