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Math Help - Linear Transformations

  1. #1
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    Linear Transformations

    Hi,
    I have a problem regarding linear transformations.
    Consider the triangle in R^2 with vertices (1,1), (3,2) and (2,3). Find a linear transformation T:R^2 to R^2 which firstly doubles the vertical distance from the origin to each of the vertices, then rotates the triangle through an angle of 5pi/6 anticlockwise about the origin, and finally reflects it across the line y=x.
    Thankyou in advance for any assistance provided.
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  2. #2
    MHF Contributor

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    Do it one step at a time. Any linear transformation on R^2 can be represented as a 2 by 2 matrix. And you want
    \begin{bmatrix} a & b \\ c & d\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix}x \\ 2y\end{bmatrix} Do that multiplication on the left, set components equal to those on the right and you can see what a and b must b.

    You could do the same thing for the rotation but it is simpler to use the fact that a counter clockwise rotation, about the origin, through \theta is represented by the matrix
    \begin{bmatrix}cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta)\end{bmatrix}

    Doing both is equivalent to multipying the matrices. Be sure you multiply in the right order: the matrix applied first on the right.
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  3. #3
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    Thankyou for your assistance halls of ivy
    I understand the technique you suggested to complete the question, its just that i'm not sure where to get started. Specifically what do i need to do given the vertices of this triangle, in order to get to the stage where i can actually start to use your technique and perform the linear transformations using matrix operations?
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