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Math Help - Affine transformation

  1. #1
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    Affine transformation

    Having a little trouble with this question:

    f is an affine transformation. The transformation f is a reflection in the line
    y=-x-2.

    By first translating an appropriate point to the origin find f in the form

    f(x)=Ax+a
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  2. #2
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    Quote Originally Posted by offahengaway and chips View Post
    Having a little trouble with this question:

    f is an affine transformation. The transformation f is a reflection in the line
    y=-x-2.

    By first translating an appropriate point to the origin find f in the form

    f(x)=Ax+a
    The line y= -x-2 is a line with slope -1 passing through the point (0, -2). The translation (x,y)-> (x,y+2) maps that point into (0, 0) and maps everypoint on the line into a point on y= -x. A reflection in that line is (x,y)->(-y, -x), which would be represented by the matrix equation A(x,y)= \begin{bmatrix}0 & -1 \\ -1 & 0\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}= \begin{bmatrix}-y \\ -x\end{bmatrix}. Finally, the translation (x,y)->(x, y-2) maps back to the original position. So we are doing A((x,y)+ (0,2))+ (0,-2)= A(x,y)+ A(0,2)+ (0,-2). A(0,2)= \begin{bmatrix}0 & -1 \\ -1 & 0\end{bmatrix}\begin{bmatrix}0 \\ 2 \end{bmatrix}= \begin{bmatrix}-2 \\ 0\end{bmatrix}

    So A(x,y)+ a is \begin{bmatrix}0 & -1 \\ -1 & 0\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}+ \begin{bmatrix}-2 \\ -2\end{bmatrix}.
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