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March 28th, 2015, 11:30 PM  #1 
Newbie Joined: Mar 2015 From: Singapore Posts: 1 Thanks: 0  Linear independence & linear dependence
Suppose that vectors v1,v2,v3 are linearly independent. (i) Prove that v1v2, v2v3 and v3v1 are linearly dependent. (ii) Prove that v1+v2, v2+v3 and v3+v1 are linearly dependent. I need you guys to help me solve this question because I'm stucked in this question for few hours already thanks in advance! 
March 29th, 2015, 03:14 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,642 Thanks: 625 
(i) Assume a(v1v2)+b(v2v3)+c(v3v1)=0. Then (ac)v1+(ba)v2+(cb)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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dependence, independence, linear 
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