Linear Independence Proof

1. Mar 28, 2006

swaldon

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?

Last edited: Mar 28, 2006
2. Mar 28, 2006

Galileo

Yes, this is sufficient. Since A is linear it should scale scalar multiples of a vector by the same factor.

3. Mar 29, 2006

Muzza

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.

4. Mar 29, 2006

HallsofIvy

Staff Emeritus
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.