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Unformatted text preview: J 2 is the matrix representing a counterclockwise rotation through the angle . But such a rotation sends any vector x into its oppositex , so that J 2 =I as you can see by direct calculation. For (b) note that xI + yJ = xy y x and uI + vJ = uv v u . By direct calculation, we nd xy y x uv v u = xuyvxvyu xv + yu xuyv = ( xuyv ) I + ( xv + yu ) J . For (c) , if ( xI + yJ ) 2 = 4 I , then x 2y 22 xy 2 xy x 2y 2 = 4 4 . Looking at the o diagonal entries, we see that either x = 0 or y = 0. But if x = 0, then the diagonal entries would bey 2 , which cannot equal 4. Hence it must be that y = 0 and x 2 = 4, or x = 2. Part (d) is almost the same: If ( xI + yJ ) 2 =4 I , then x 2y 22 xy 2 xy x 2y 2 = 44 . 21 /september/ 2005; 22:09 23...
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This note was uploaded on 10/03/2010 for the course MATH 380 taught by Professor Gladue during the Fall '08 term at Roger Williams.
 Fall '08
 Gladue
 Calculus, Matrices

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