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Linear Transformation of Matrix

  1. Apr 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Let A[itex]_{2x2}[/itex] have all entries=1 and let T: M[itex]_{2x2}[/itex][itex]\rightarrow[/itex]M[itex]_{2x2}[/itex] be the linear transformation defined by T(B)=AB for all B[itex]\in[/itex]M[itex]_{2x2}[/itex]

    Find the matrix C=[T]s,s, where S is the standard basis for M[itex]_{2x2}[/itex]

    My solution:

    Standard basis for M[itex]_{2x2}[/itex]={(1,0),(0,1)}
    T(1,0)=(1,1)
    T(0,1)=(1,1)
    [T]s,s=(1,1;1,1)

    I'm not sure how correct this is. Any advice would be appreciated.
     
  2. jcsd
  3. Apr 17, 2012 #2
    Still confused on this one
     
  4. Apr 17, 2012 #3

    Dick

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    The standard basis for M_2x2 is four matrices [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[1,0]] and [[0,0],[0,1]]. You can express any other matrix as a linear sum of those. Now take a look at the problem again.
     
  5. Apr 17, 2012 #4
    Doesn't this still produce the same vector?
     
  6. Apr 17, 2012 #5

    Dick

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    What vector? What is A times the first basis matrix?
     
  7. Apr 17, 2012 #6
    Sorry, the same matrix

    A times first basis matrix is

    [[1,0][0,0]]
    then
    [[0,0][1,0]], [[0,1][0,0]], and [[0,0][0,1]]
     
  8. Apr 17, 2012 #7

    Dick

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    [[1,1],[1,1]]*[[1,0],[0,0]] isn't equal to [[1,0],[0,0]].
     
  9. Apr 17, 2012 #8
    It appears it isn't

    [[1,0][1,0]]

    then

    [[0,1][0,1]], [[1,0][1,0]], and [[0,1][0,1]]
     
  10. Apr 17, 2012 #9

    Dick

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    Ok, so work on what the matrix C should be. It should be 4x4 since you have four basis elements.
     
  11. Apr 17, 2012 #10
    Would this be

    [[1,0,0,1],[1,0,0,1],[1,0,0,1],[1,0,0,1]]?
     
  12. Apr 17, 2012 #11

    Dick

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    It would depend on which column represents which basis element. You should spell that out. But no I don't think that's it. How did you conclude that?
     
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