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Math Help - linear transformation's effect on unit square

  1. #1
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    linear transformation's effect on unit square

    The linear transformation is given as T \begin{bmatrix}x \\ y \end{bmatrix} = \begin{bmatrix}\ x+y \\ x-y\end{bmatrix} and I have to show the effect on bothe the unit square as well as a triangle with vertices (-1,0); (0,1); (1,0).

    As far as it's effect on the unit circle would it simply a reflection around the x-axis (to account for the -y in x-y) as well as a shift up by y in the x direction? ( I know that doesn't quite make sense..)

    For the triangle I'm even more confused, would it still be a refection over the x-axis, but would it also lead to some sort of horizontal shear?
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  2. #2
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    Re: linear transformation's effect on unit square

    Check the vertices. I will help out for the triangle.
    T\begin{bmatrix}-1 \\ 0\end{bmatrix} = \begin{bmatrix}-1+0 \\ -1-0\end{bmatrix} = \begin{bmatrix}-1 \\ -1\end{bmatrix} (shifted down one unit).
    T\begin{bmatrix}0 \\ 1\end{bmatrix} = \begin{bmatrix}0+1 \\ 0-1\end{bmatrix} = \begin{bmatrix}1 \\ -1\end{bmatrix} (shifted down two units and to the right one unit).
    T\begin{bmatrix}1 \\ 0\end{bmatrix} = \begin{bmatrix}1+0 \\ 1-0\end{bmatrix} = \begin{bmatrix}1 \\ 1\end{bmatrix} (shifted up one unit).

    The line segment between (-1,0) and (1,0) is given by the line y=0. So that edge is going to:
    T\begin{bmatrix}x \\ 0\end{bmatrix} = \begin{bmatrix}x+0 \\ x-0\end{bmatrix} = \begin{bmatrix}x \\ x\end{bmatrix} which is the line y = x, so a slanted line between (-1,-1) and (1,1).

    The line segment between (-1,0) and (0,1) is given by the line y=x. So that edge is going to:
    T\begin{bmatrix}x \\ x\end{bmatrix} = \begin{bmatrix}x+x \\ x-x\end{bmatrix} = \begin{bmatrix}2x \\ 0\end{bmatrix} So that edge is now horizontal going from (-1,-1) to (1,-1).

    The line segment between (0,1) and (1,0) is given by the line y = -x. So that edge is going to:
    T\begin{bmatrix}x \\ -x\end{bmatrix} = \begin{bmatrix}x + (-x) \\ x - (-x)\end{bmatrix} = \begin{bmatrix}0 \\ 2x\end{bmatrix} so that edge is now a vertical line going from (1,-1) to (1,1). In other words, the triangle is rotated 225 degrees = \dfrac{5\pi}{4} radians and stretched by a factor of \sqrt{2}.

    Consider the unit circle. Every point of the unit circle is given by (\cos \theta,\sin \theta). So,
    T\begin{bmatrix}\cos \theta \\ \sin \theta\end{bmatrix} = \begin{bmatrix}\cos \theta + \sin \theta \\ \cos \theta - \sin \theta\end{bmatrix}

    Now, points on the unit sphere have the property that x^2+y^2 = 1. Now,
    \begin{align*}(\cos \theta + \sin \theta)^2 + (\cos \theta - \sin \theta)^2 & = \cos^2\theta + 2\sin \theta \cos \theta + \sin^2\theta + \cos^2\theta - 2\sin\theta \cos\theta + \sin^2\theta \\ & = 2(\cos^2\theta + \sin^2\theta) \\ & = 2\end{align*}

    So, you now have a circle of radius \sqrt{2}. Seeing a pattern?
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