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Thread: Linear Transformations

  1. #1
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    Linear Transformations

    let e1 = $\displaystyle
    \begin{bmatrix}1 \\ 0 \end{bmatrix}
    $, e2 =$\displaystyle
    \begin{bmatrix}0 \\ 1 \end{bmatrix}
    $, y1= $\displaystyle
    \begin{bmatrix}2 \\ 5 \end{bmatrix}
    $,y2 = $\displaystyle
    \begin{bmatrix}-1 \\ 6 \end{bmatrix}
    $


    and T: R^2 --> R^2 be a linear transformation that maps e1 into y1 and maps e2 into y2.

    find the images of $\displaystyle
    \begin{bmatrix}5 \\ -3 \end{bmatrix}
    $ and $\displaystyle
    \begin{bmatrix}x1 \\ x2\end{bmatrix}
    $


    Ive been messing with this for awhile, I simply dont understand what its asking, or what I am supposed to do.

    help!
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  2. #2
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    You are given that $\displaystyle
    T\left( {\left[ \begin{gathered}
    1 \hfill \\
    0 \hfill \\
    \end{gathered} \right]} \right) = \left[ \begin{gathered}
    2 \hfill \\
    5 \hfill \\
    \end{gathered} \right]\,\& \,T\left( {\left[ \begin{gathered}
    0 \hfill \\
    1 \hfill \\
    \end{gathered} \right]} \right) = \left[ \begin{gathered}
    - 1 \hfill \\
    6 \hfill \\
    \end{gathered} \right]$.
    You know that $\displaystyle \left[ \begin{gathered}
    5 \hfill \\
    - 3 \hfill \\
    \end{gathered} \right] = 5\left[ \begin{gathered}
    1 \hfill \\
    0 \hfill \\
    \end{gathered} \right] - 3\left[ \begin{gathered}
    0 \hfill \\
    1 \hfill \\
    \end{gathered} \right]$
    Now apply the linear transformation.
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  3. #3
    Member
    Joined
    Jan 2008
    Posts
    175
    Quote Originally Posted by Plato View Post
    You are given that $\displaystyle
    T\left( {\left[ \begin{gathered}
    1 \hfill \\
    0 \hfill \\
    \end{gathered} \right]} \right) = \left[ \begin{gathered}
    2 \hfill \\
    5 \hfill \\
    \end{gathered} \right]\,\& \,T\left( {\left[ \begin{gathered}
    0 \hfill \\
    1 \hfill \\
    \end{gathered} \right]} \right) = \left[ \begin{gathered}
    - 1 \hfill \\
    6 \hfill \\
    \end{gathered} \right]$.
    You know that $\displaystyle \left[ \begin{gathered}
    5 \hfill \\
    - 3 \hfill \\
    \end{gathered} \right] = 5\left[ \begin{gathered}
    1 \hfill \\
    0 \hfill \\
    \end{gathered} \right] - 3\left[ \begin{gathered}
    0 \hfill \\
    1 \hfill \\
    \end{gathered} \right]$
    Now apply the linear transformation.
    got it!
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