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Linear Algebra - How to represent this transformation?

  1. Feb 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Given that T is the linear operator on R3 with T(1,1,1) = (0,0,1), T(1,1,0) = (1,2,1), T(1,0,0) = (0,-1,0), determine the eigenvalues of T and a corresponding eigenvector for each eigenvalue.


    2. Relevant equations



    3. The attempt at a solution

    I know how to find eigenvalues and vectors but usually the matrix is given.. I'm not sure how to represent these transformations as one whole matrix.. or do I need to do them separately?
     
  2. jcsd
  3. Feb 16, 2010 #2

    vela

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    You can get the columns of the matrix by applying T to the basis vectors, so you just need to figure out how to calculate T(1,0,0), T(0,1,0), and T(0,0,1) to find the matrix.
     
  4. Feb 16, 2010 #3
    Okay in that case I can make an augmented matrix with the given relations and row reduce the left side to be the standard basis and the augmented side will be affect of T on the standard basis? Then I can transpose that to get the matrix?
     
    Last edited: Feb 16, 2010
  5. Feb 16, 2010 #4

    vela

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    If you can find a, b, and c such that

    (0,1,0) = a(1,1,1)+b(1,1,0)+c(1,0,0)

    then

    T(0,1,0)=T[a(1,1,1)+b(1,1,0)+c(1,0,0)]=...

    Use T's linearity to evaluate the RHS. Do the same for the vector (0,0,1).
     
  6. Feb 16, 2010 #5
    I don't really understand.. How can I know what T does just by looking at the vectors that T is applied to..?
     
  7. Feb 16, 2010 #6

    vela

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    Like I said, use the fact that T is linear to evaluate the RHS.
     
  8. Feb 16, 2010 #7
    Okay so for
    (0,1,0) a = 0, b= 1, c=-1
    (0,0,1) a = 1, b = -1, c = 0
     
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