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Linear Algebra - Change of basis question

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

    Let A = E4 in R4 (standard basis) and B = {x^2, x, 1} in P2 over R. If T is the linear transformation that is represented by
    [tex]

    \begin{bmatrix}1 & 1 & 0 & 1\\0 & 0 & 1 & -1\\1 & 1 & 0 & 1 \end{bmatrix}

    [/tex]

    relative to A and B, find the matrix that represents T with respect to A' and B' where
    A' = {(1,0,0,0), (0,0,1,0), (1,-1,0,0), (0,-1,1,1)}
    B' = {x^2 + 1, x, 1}

    2. Relevant equations



    3. The attempt at a solution

    So by looking at this matrix T, it's clear that its a transformation from A to B, so we want the transformation matrix [tex]T_{B'A'}[/tex],
    which is: [tex]T_{B'A'} = I_{B'B}T_{BA}I_{AA'}[/tex]

    So I need to find [tex]I_{AA'}[/tex] and [tex] I_{B'B}[/tex].

    For [tex]I_{AA'}[/tex], I write A' wrt A(which is standard basis of R4):

    I get : [tex]I_{AA'} = \begin{bmatrix}1 & 0 & 1 & 0 \\0 & 0 & -1 &-1\\0 & 1 & 0 &1\\ 0&0&0&1 \end{bmatrix} [/tex]

    Then for [tex] I_{B'B}[/tex], I write B wrt B', and get

    [tex]I_{B'B} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\-1 & 0 & 1 \end{bmatrix} [/tex]

    Now I put them together to get something with lots of zeros.. which doesn't seem right?
     
  2. jcsd
  3. Feb 1, 2010 #2

    vela

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    Your answer is correct. (At least it sounds correct.) You have the representations of the A' basis in the natural basis, so try applying them to the given T and see what you get. Then look at how those results would be represented in the B' basis. You'll see why the transformed T has so many zeros.
     
  4. Feb 2, 2010 #3
    When I put them all together I get:
    [tex]
    \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\-1 & 0 & 1 \end{bmatrix} \begin{bmatrix}1 & 1 & 0 & 1\\0 & 0 & 1 & -1\\1 & 1 & 0 & 1 \end{bmatrix} \begin{bmatrix}1 & 0 & 1 & 0 \\0 & 0 & -1 &-1\\0 & 1 & 0 &1\\ 0&0&0&1 \end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 0 & 0 \end{bmatrix}
    [/tex] ??
     
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