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Linear Transformations and Image of a Matrix

  1. Feb 18, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider a 2x2 matrix A with A2=A.
    If vector w is in the image of A, what is the relationship between w and Aw?

    2. Relevant equations
    Linear transformation T(x)=Ax
    Image of a matrix is the span of its column vectors

    3. The attempt at a solution
    I know that vector w is one of the column vectors of A, seeing as it is in the image of A.
    At first, I tried to work out Aw on paper by substituting [x1 x2] for the second column vector in A, since it isn't specified.
    I got the following four equations:
    w1 = w1^2+x1w2
    w2=w1w2+x2w2
    x1=w1x1+x1x2
    x2=x1w2+w2^2
    However, I couldn't get anything from there.

    Now, I'm trying to think of it more conceptually (I believe the point of the problem is to think about it more abstractly), but I'm not sure which direction to head in. All I know is that Ax can equal w, so Aw may also equal w, but the problem doesn't say that w IS the image of A, just that it is IN the image of A.

    Any suggestions on how to think about this?
     
  2. jcsd
  3. Feb 18, 2015 #2

    Dick

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    Science Advisor
    Homework Helper

    Asking "what is the relationship" is a pretty ambiguous question. I'd think about it more conceptually this way: A(Aw)=A^2(w)=Aw. What is A(Aw-w)? What do those tell you conceptually? Here's a hint about what they might be fishing for, look up projection operator. That's my best guess.
     
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