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Linear Transformation

  1. Sep 4, 2012 #1
    Linear transformation [itex]T:\,\mathbb{R}^3\,\to\,\mathbb{R}^4[/itex]

    Find the standard matrix A for T

    [tex]
    T\left(x_1,x_2,x_3\right)\,=\,\left(x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3\right)
    [/tex]

    [tex]
    \mathbf{v}\,=\,\begin{bmatrix}
    x_1\\
    x_2\\
    x_3
    \end{bmatrix}\,\,\,\,T\left(\mathbf{v}\right)\,=\,T\,\begin{bmatrix}
    x_1\\
    x_2\\
    x_3
    \end{bmatrix}\,=\,\begin{bmatrix}
    x_1 + x_2 + x_3\\
    x_2 + x_3\\
    3x_1 + x_2\\
    2x_2 + x_3
    \end{bmatrix}\,=\,\begin{bmatrix}
    1 & 1 & 1\\
    0 & 1 & 1\\
    3 & 1 & 0\\
    0 & 2 & 1
    \end{bmatrix}
    [/tex]


    Is that correct? I feel like I'm just walking down a blind alley with this problem. The text is sorta convoluted in this section and I can't seem to find any supplementary material that I feel is helpful.

    Thanks.
     
  2. jcsd
  3. Sep 4, 2012 #2

    LCKurtz

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    Your last = sign has a 4 by 1 matrix equal to a 4 by 3 matrix. You left something out. There doesn't seem to be a question in your post but once you fix that last matrix equality your work looks correct.
     
  4. Sep 4, 2012 #3
    That's what I'm really confused about. How do you show the transformation with the matrices?
     
  5. Sep 4, 2012 #4

    LCKurtz

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    You just need the matrix $$
    X = \begin{bmatrix}
    x_1\\ x_2\\ x_3
    \end{bmatrix}$$

    column matrix on the right of that last matrix. Then, calling your matrix ##A## you have ##T(X) = AX##
     
  6. Sep 4, 2012 #5
    Okay, thank you.
     
  7. Sep 4, 2012 #6

    Ray Vickson

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    It would have been correct if you had written
    [tex] T(\mathbf{v}) = \begin{bmatrix}
    1 & 1 & 1\\
    0 & 1 & 1\\
    3 & 1 & 0\\
    0 & 2 & 1
    \end{bmatrix} \begin{bmatrix}
    x_1\\
    x_2\\
    x_3
    \end{bmatrix}
    [/tex]

    RGV
     
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