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matrices-as-maps

# matrices-as-maps - Math 260 Spring 2012 Jerry L Kazdan...

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Math 260, Spring 2012 Jerry L. Kazdan Matrices as Maps We now discuss viewing systems of equations as maps. Think of this an an introduction to computer graphics . We’ll use these ideas throughout Math 260. 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 ax 1 + bx 2 = y 1 cx 1 + dx 2 = y 2 (1) 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 = 0, then y 1 = a and y 2 = c . Thus the point (1 , 0) is mapped to the point ( a, c ). Similarly, the point (0 , 1) is mapped to ( b, d ). Since the coeﬃcients in these equations contain all the useful information, it is useful to collect them in a box as a matrix A := ± a b c d ² , and also write x := ± x 1 x 2 ² and y := ± y 1 y 2 ² . Now write the equations (1) using the shorthand

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matrices-as-maps - Math 260 Spring 2012 Jerry L Kazdan...

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