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March 28th, 2015, 10:30 PM   #1
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Linear independence & linear dependence

Suppose that vectors v1,v2,v3 are linearly independent.
(i) Prove that v1-v2, v2-v3 and v3-v1 are linearly dependent.
(ii) Prove that v1+v2, v2+v3 and v3+v1 are linearly dependent.

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March 29th, 2015, 02:14 PM   #2
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(i) Assume a(v1-v2)+b(v2-v3)+c(v3-v1)=0.
Then (a-c)v1+(b-a)v2+(c-b)v3=0.
Linear independence forces a=b=c, so the difference vectors are dependent, using the same coefficient for each.

(ii)When you work with the sum vectors, you end up with a=-c, b=-a, c=-b, which forces a=b=c=0, so the sum vectors are independent.
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