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Math Help - Linear Transformation

  1. #1
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    Linear Transformation

    How would you find the linear transformation that takes each (x,y) point and projects onto the line y=x?
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  2. #2
    A Plied Mathematician
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    How do you want points to travel to the line y = x? I could easily think of a way to do this using rotation matrices. But then a point would not travel to the closest point on y = x from its original location. It would travel to the nearest point on y = x that is on the circle with radius equal to its own distance from the origin. In other words, they would travel around on a circle until they reached y = x. But if you use the term "projection", I think you mean that each point goes straight towards the nearest point on y = x, right?
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  3. #3
    MHF Contributor

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    I would interpret "projection" as meaning "orthogonal projection". That is, to project (x_0, y_0) onto the line y= x, draw the line throught (x_0, y_0) perpendicular to the line y= x. The projection is at the intersection of those two lines.

    Of course, any line perpendicular to y= x is of the form y= -x+ A. Taking y= y_0, x= x_0, We have y_0= -x_0+ A so that A= x_0+ y+0. The line perpendicular to y= x is y= -x+ x_0+ y_0. That intersects the line y= x when x= -x+ x_0+ y_0 or 2x= x_0+ y_0 so that x= \frac{x_0+ y_0}{2}.

    The point (x_0, y_0) is projected onto the line y= x at the point \left(\frac{x_0+ y_0}{2},\frac{x_0+ y_0}{2}\right).

    That is L((x_0, y_0))= \left(\frac{x_0+ y_0}{2}, \frac{x_0+ y_0}{2}\right)

    You could also write this as the matrix multiplication
    \begin{bmatrix}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}\begin{bmatrix}x_0 \\ y_0\end{bmatrix}
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