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Represent a matrix on the (x,y)-plane

  1. Oct 2, 2005 #1

    hgj

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    I'm having trouble understanding how to represent a matrix on the (x,y)-plane. It seems like my classes expect me to know this, but I was never shown how. I've heard that there are a lot of connections between matrices and geometry, that matrices can be used to provide a geometric representation of something, but I don't understand how. For example, if I have a 2x2 matrix [tex]\left(\begin{array}{cc}x&y\\0&1\end{array}\right)[/tex], how does that look in the (x,y)-plane? I really want to understand this better.
     
  2. jcsd
  3. Oct 2, 2005 #2
    Well you could look at it as two vectors.
     
  4. Oct 2, 2005 #3

    HallsofIvy

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    Staff Emeritus
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    I have no idea what you mean by "represent a matrix on the xy-plane".

    Perhaps you mean "represent a matrix as a linear transformation on the xy-plane.

    If you are going to do that, it would be better not to have "x" and "y" in the matrix itself.
    If the matrix were [tex]\left(\begin{array}{cc}2&3\\0&1\end{array}\right)[/tex]
    for example then we could note that multiplying the "point" (x,y) by it gives
    [tex]\left(\begin{array}{cc}2&3\\0&1\end{array}\right)\left(\begin{array}{cc}x\\y\end{array}\right)= \left(\begin{array}{cc}2x+3y\\y\end{array}\right)[/tex]

    You could also, perhaps more simply, apply it to the "basis" vectors (1, 0) and (0,1) and see what happens there:

    [tex]\left(\begin{array}{cc}2&3\\0&1\end{array}\right)\left(\begin{array}{cc}1\\0\end{array}\right)= \left(\begin{array}{cc}2\\0\end{array}\right)[/tex]

    [tex]\left(\begin{array}{cc}2&3\\0&1\end{array}\right)\left(\begin{array}{cc}0\\1\end{array}\right)= \left(\begin{array}{cc}3\\1\end{array}\right)[/tex]

    Draw the lines through (0,0) and each of those and imagine them as what the matrix does to the xy-axes. Of course, all points between the xy-axes are changed into points between those lines.
     
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