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Linear Independence Proof

  1. Mar 28, 2006 #1
    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. jcsd
  3. Mar 28, 2006 #2

    Galileo

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    Yes, this is sufficient. Since A is linear it should scale scalar multiples of a vector by the same factor.
     
  4. Mar 29, 2006 #3
    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.
     
  5. Mar 29, 2006 #4

    HallsofIvy

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    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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