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symmetries

# symmetries - Math 260 Spring 2012 Jerry L Kazdan...

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Math 260, Spring 2012 Jerry L. Kazdan Representing Symmetries by Matrices If order to understand and work with the symmetries of an object (the symmetries of a square is a simple example), one would like a way to compute, not just wave your hands. For an object with complicated symmetries, this is essential. The standard technique goes back to Descartes’ introduction of coordinates in geometry. Say one has two copies of the plane, the first with coordinates ( x 1 , x 2 ), the second with coordinates ( y 1 , y 2 ). Then the high school equations x 1 + 2 x 2 = y 1 (1) x 1 x 2 = y 2 (2) can be thought of as a mapping from the ( x 1 , x 2 ) plane to the ( y 1 , y 2 ) plane. For instance, if x 1 = 1 and x 2 = 1, then y 1 = 1 and y 2 = 2. Thus the point (1 , 1) is mapped to the point ( 1 , 2). Similarly, the equations 0 x 1 2 x 2 = y 1 (3) x 1 + 0 x 2 = y 2 (4) defines a map that takes (1 , 0) to (0 , 1) and (0 , 1) to ( 2 , 0). It can be thought of as a vertical stretching by a factor of 2 followed by a counter-clockwise rotation by 90 degrees.

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symmetries - Math 260 Spring 2012 Jerry L Kazdan...

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