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Math Help - Matrix transformation

  1. #1
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    Matrix transformation

    Consider the linear transformation, T:\mathbb{R}^2\rightarrow \mathbb{R}^2, for which T(x) is found by rotating the point x through an angle of \frac{2\pi}{3} anticlockwise about the origin.

    a) Find the 2 x 2 matrix A for which T(x) = Ax

    I wasn't quite sure how T(x) could = Ax and as such I wasn't entirely sure how to answer the question. I just applied the same method I had been using for all the previous question to get:

     \left(\begin{matrix} cos(\frac{2\pi}{3}) &-sin(\frac{2\pi}{3}) \\sin(\frac{2\pi}{3}) & cos(\frac{2\pi}{3})\end{matrix}\right)

    Is this the correct answer for matrix A? I thought this would be T(x)? What is the x value that A is multiplied by?

    b) Write down the transformation that returns the point x to its original position.

    Since we rotated everything \frac{2\pi}{3} counter-clockwise, I would presume that to reverse that transformation we would rotate everything by -\frac{2\pi}{3} to get:

     \left(\begin{matrix} cos(\frac{2\pi}{3}) & sin(\frac{2\pi}{3}) \\-sin(\frac{2\pi}{3}) & cos(\frac{2\pi}{3})\end{matrix}\right)

    Is this correct?

    Thanks.
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  2. #2
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    Tejas
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    Re: Matrix transformation

    normally an element x of \mathbb{R}^2 is written as (x,y) (sometimes as (x_1,x_2)). so for the example here:

    T(x,y) = (\cos(2\pi/3)x+\sin(2\pi/3)y, \cos(2\pi/3)y-\sin(2\pi/3)x)

    you can check your answer by multiplying your two matrices together, and verifying you get the 2x2 identity matrix.
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