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
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-> 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
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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
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.
