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Homework Help: Matrix of linear transformation

  1. Apr 1, 2010 #1
    1. The problem statement, all variables and given/known dataFind the matrix of the transformation:

    [tex]T: R^{2} \rightarrow R^{2x2}[/tex]

    [tex]
    \[
    T(a,b) =
    \left[ {\begin{array}{cc}
    a & 0 \\
    0 & b \\
    \end{array} } \right]
    \][/tex]



    2. Relevant equations



    3. The attempt at a solution
    I choose the standard bases for [tex]R^{2}[/tex] and [tex]R^{2x2}[/tex] and call them b and b' respectively.

    [tex]T(1,0) = 1e_{1} + 0e_{2} + 0e_{3} + 0e_{4}[/tex]
    [tex]T(0,1) = 0e_{1} + 0e_{2} + 0e_{3} + 1e_{4}[/tex]

    This gives me a matrix of

    [tex]
    \[
    \left[ {\begin{array}{cc}
    1 & 0 \\
    0 & 0 \\
    0 & 0 \\
    0 & 1 \\
    \end{array} } \right]
    \][/tex]

    However, this doesn't work when I multiply by the column vector (a,b). I get a column vector of (a, 0, 0, b) instead of a 2x2 matrix. What's going on?
     
  2. jcsd
  3. Apr 1, 2010 #2

    radou

    User Avatar
    Homework Helper

    If your R^2x2 is the subspace of all 4x4 matrices of this given form, what is its dimension? You have too many basis vectors here.
     
  4. Apr 1, 2010 #3
    R^2x2 is the space of all 2x2 matrices. It has 4 basis vectors with 1 in the i,j position and zeroes everywhere else.
     
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