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Math Help - Linear transformation

  1. #1
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    Exclamation Linear transformation

    Hello! I have som problems with this one. maybe someone can help me?

    Find the standard matrix for the linear transformation.

    T: R^2 --> R^2 dilates a vector by a factor of 3, then reflects that vector about the line y=x, and then projects that vector orthogonally onto the y-axis.

    I am supposed to use this theorem while solving :

    "Let T: R^n --> R^m be a linear transformation, and suppose that vectors are expressed in column form.
    If e1, e2,...,en are the standard vectors in R^n, and if x is any vector in R^n, then T(x) can be expressed as

    T(x)=Ax where A = [T(e1) T(e2) ... T(en)]"


    =)
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  2. #2
    Super Member redsoxfan325's Avatar
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    Swampscott, MA
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    Quote Originally Posted by Fredrik373 View Post
    Hello! I have som problems with this one. maybe someone can help me?

    Find the standard matrix for the linear transformation.

    T: R^2 --> R^2 dilates a vector by a factor of 3, then reflects that vector about the line y=x, and then projects that vector orthogonally onto the y-axis.

    I am supposed to use this theorem while solving :

    "Let T: R^n --> R^m be a linear transformation, and suppose that vectors are expressed in column form.
    If e1, e2,...,en are the standard vectors in R^n, and if x is any vector in R^n, then T(x) can be expressed as

    T(x)=Ax where A = [T(e1) T(e2) ... T(en)]"


    =)
    First it dilates by 3:

    \left[\begin{array}{cc}3&0\\0&3\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]

    Then it reflects about y=x:

    \left[\begin{array}{cc}0&1\\1&0\end{array}\right]\left[\begin{array}{cc}3&0\\0&3\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]

    Then it projects orthogonally onto the y-axis:

    \left[\begin{array}{cc}0&0\\0&1\end{array}\right]\left[\begin{array}{cc}0&1\\1&0\end{array}\right]\left[\begin{array}{cc}3&0\\0&3\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]

    Putting it all together gives us:

    \left[\begin{array}{cc}0&0\\3&0\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]
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  3. #3
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    Thank you very much! =)
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