SOLUTION: I was given a homework sheet in class and have no idea what chapter it is from. Let A be an nxn matrix and suppose that v1 and v2 are nonzero vectors in R^n that satisfy {{{Av1=v1

Question 31686This question is from textbook Elementary Linear Algebra with Applications
: I was given a homework sheet in class and have no idea what chapter it is from. Let A be an nxn matrix and suppose that v1 and v2 are nonzero vectors in R^n that satisfy and . Prove that the set {v1, v2} is linearly independent.
Thank you for your help. This question is from textbook Elementary Linear Algebra with Applications

Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
LET US PROVE THIS BY REDUXIO-AD-ABSURDUM OR BY CONTRADICTION.
LET US ASSUME THAT V1 AND V2 ARE NOT LINEARLY INDEPENDENT.
THAT IS THEY ARE LINEARLY DEPENDENT.
SO LET V1=KV2 WHERE K IS A SCALAR.
WE ARE GIVEN
AV1=V1 ........AKV2=V1....OR......KAV2=V1
BUT AV2=2V2
SO K*2V2=V1
2KV2=V1=2KV2...BUT OUR ASSUMPTION IS V1=KV2
HENCE KV2=2KV2...OR....K=2K...OR....K=0
SO...V1=0V2=0...BUT V1 IS NONZERO VECTOR AS PER HYPOTHESIS.
HENCE THIS IS NOT CORRECT.SO OUR ASSUMPTION THAT V1 AND V2 ARE LINEARLY DEPENDENT IS WRONG.HENCE V1 AND V2 ARE LINEARLY INDEPENDENT.
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