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Linear Transformation - The Matrix of (not so hard)

  1. Jan 15, 2007 #1
    1. The problem statement, all variables and given/known data

    I have a linear map T:M(2x2) -------> M(2x2) defined by T(B) = [2 3; 4 0] * B

    Find a 4 × 4 matrix representation of this linear transformation with respect to the basis of M(2×2)

    2. Relevant equations

    T(B) = [2 3; 4 0] * B


    and the basis for M(2X2) is:

    [1 0; 0 0]
    [0 1; 0 0]
    [0 0; 1 0]
    [0 0; 0 1]



    3. The attempt at a solution


    T[1 0; 0 0] = [2 3; 4 0]*[1 0; 0 0] = [2 4; 0 0]
    T[0 1; 0 0] = [2 3; 4 0]*[0 1; 0 0] = [0 0; 2 4]
    T[0 0; 1 0] = [2 3; 4 0]*[0 0; 1 0] = [3 0; 0 0]
    T[0 0; 0 1] = [2 3; 4 0]*[0 0; 0 1] = [0 3; 0 0]

    Therefore, the matrix would be:

    [2 0 3 0;
    4 0 0 3;
    0 2 0 0;
    0 4 0 0]

    Could somebody please veryify this for me.

    I'd appreciate that
     
  2. jcsd
  3. Jan 15, 2007 #2
    As long as you put the image vectors as columns in the 4x4 matrix, and you multiplied the 2x2 matrices correctly (double check this), you should not be worrying so much.
     
    Last edited: Jan 15, 2007
  4. Jan 15, 2007 #3
    Oh...

    So what you are saying is that I DID my procedure correctly....and that final answer would be correct as long as my aritmatic is correct? :smile:

    That makes me feel better...
     
  5. Jan 15, 2007 #4
    that's right but double check the products wrt to the ordered basis you have chosen.

    whoa, that's it. my dad is kicking me out of the computer for making too many posts.
     
    Last edited: Jan 15, 2007
  6. Jan 15, 2007 #5
    Ohh okay...thanks.

    And that is funny lol.
     
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