Linear Independence Proof

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?

Galileo

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

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.

HallsofIvy

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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swaldon	Linear Independence Proof 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 AV1=V1 and AV2 = 2*V2. Is this enough information, or is more needed to prove the LI of the 2 vectors?

Galileo Recognitions: Homework Help Science Advisor	Yes, this is sufficient. Since A is linear it should scale scalar multiples of a vector by the same factor.

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.

HallsofIvy Recognitions: Gold Member Science Advisor Staff Emeritus	Linear Independence Proof 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.