Find the matrix of the following...

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Department of Mathematics, University of Toronto MAT223H1S - Linear Algebra I Winter 2016 Tutorial Problems 5 1. Find the matrix of the following reflections: (i) R 1 : R 2 R 2 , is a reflection in the line x 1 - 3 x 2 = 0. (ii) R 2 : R 2 R 2 is a reflection in the line 2 x 1 = x 2 . (a) Find the matrix of the composition R 1 R 2 : R 2 R 2 and show that it can be identified as a rotation and determine the angle of rotation. (b) Sketch, as accurately as possible, the image of the three unit squares pictured below under each of the transformations R 1 , R 2 , and R 1 R 2 . - 2 - 1 1 2 3 4 1 2 3
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(i) For the line x 1 - 3 x 2 = 0, where n = 1 - 3 , we compute using the above formula that R 1 1 0 = 1 0 - 2 1 10 1 - 3 = 4 / 5 3 / 5 , R 1 0 1 = 0 1 - 2 - 3 10 1 - 3 = 3 / 5 - 4 / 5 So, the corresponding matrix is A = 4 / 5 3 / 5 3 / 5 - 4 / 5 . (ii) Similarly, for the line 2 x 1 - x 2 = 0 we have n = 2 - 1 , and we find R 2 1 0 = - 3 / 5 4 / 5 , R 2 0 1 = 4 / 5 3 / 5 The corresponding matrix is then B = - 3 / 5 4 / 5 4 / 5 3 / 5 . (a) The matrix of the composition R 1 R 2 is the product AB = 4 / 5 3 / 5 3 / 5 - 4 / 5 - 3 / 5 4 / 5 4 / 5 3 / 5 = 0 1 - 1 0 Recall that the matrix corresponding to a counter-clockwise rotation in angle θ is cos θ - sin θ sin θ cos θ . Substituting θ = 3 π 2 , we get precisely 0 1 - 1 0 . Hence R 1 R 2 is a counter-clockwise rotation in angle 3 π 2 .
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