Can we prove linear independence with just matrix and vector information?

swaldon
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Is it possible to prove 2 vectors are linearly independent with just the following information?:

A is an nxn matrix. V1 and V2 are non-zero vectors in Rn such that A*V1=V1 and A*V2 = 2*V2.

Is this enough information, or is more needed to prove the LI of the 2 vectors?
 
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Yes, this is sufficient. Since A is linear it should scale scalar multiples of a vector by the same factor.
 
This is a special case of a more general theorem that states that any set of eigenvectors of a matrix (linear transformation) are linearly independent if the eigenvectors "belong" to different eigenvalues.
 
Specifically, suppose CV1+ DV2= 0 and apply A to both sides: A(CV1+ DV2)= CAV1+ DAV2= CV1+ 2DV2= 0. Now subtract the first equation from that one.
 

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