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Linear Transformations - formula

  1. Jul 24, 2005 #1
    Hello. I am given the following:

    T([1,2,-3]) = [1,0,4,2]
    T([3,5,2]) = [-8,3,0,1]
    T([-2,-3,-4]) = [0,2,-1,0]

    And of course I know that:

    T(x) = Ax

    and I want to find the matrix A.

    So, from the individual equations, I construct:

    A[1, 2, -3] = [1, 0, 4, 2] (please forgive, these are actually col. vectors)

    I do something similar for the other two, and come up with the equation below. I can solve this equation to obtain (the correct) matrix A, but I can't seem to find an explanation for why it is possible to throw it all together into "combined matrices." Could anyone help? Thanks!

    [tex]
    A\begin{pmatrix}
    1 & 3 & -2\\
    2 & 5 & -3\\
    -3 & 2 & -4\end{pmatrix} =
    \begin{pmatrix}
    1 & -8 & 0\\
    0 & 3 & 2\\
    4 & 0 & -1\\
    2 & 1 & 0\end{pmatrix}
    [/tex]
     
  2. jcsd
  3. Jul 25, 2005 #2

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    Because the last matrix is equivalent to the three individual equations. This just follows from matrix multiplication. Schematically, if the v_n's are column vectors, you can write:

    [tex]A\left(
    \begin{array}{cccc}
    \vert & \vert & \vert & \vdots \\
    v_1 & v_2 & v_3 & \vdots \\
    \vert & \vert & \vert & \vdots \\
    \end{array}\right)=\left(
    \begin{array}{cccc}
    \vert & \vert & \vert & \vdots \\
    Av_1 & Av_2 & Av_3 & \vdots \\
    \vert & \vert & \vert & \vdots \\
    \end{array}\right)[/tex]
     
  4. Jul 25, 2005 #3
    Got it. Thanks a lot!
     
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