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## Homework Statement

Let(v

_{1}, v

_{2}, v

_{3}) be three linearly independent vectors in a vector space V. Is the set {v

_{1}-v

_{2}, v

_{2}-v

_{3}, v

_{3}-v

_{1}} linearly dependent or independent?

## Homework Equations

Linearly independent is when c1v

_{1}+c2v

_{2}+...+ckv

_{k}=0

and c1=c2=...ck=0

## The Attempt at a Solution

c1(v

_{1}-v

_{2})+ c2(v

_{2}-v

_{3})+ c3(v

_{3}-v

_{1})=0

c1-c3=0

-c1+c2=0

-c2+c3=0

therefore c1=c2=c3 and since c1, c2 and c3 are zero because for the first set of independent vectors I got c1v

_{1}+c2v

_{2}+c3v

_{3}=0 all c1=c2=c3=0,

which means this is the case for the second set and it must be linearly independent.

This is what i got but my answer key says the second set is linearly dependent. I'm having trouble seeing why.

Thanks for any help