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Unformatted text preview: The University of Sydney Math3061 Geometry and Topology Web page: www.maths.usyd.edu.au/u/UG/SM/MATH3061/ 2009 Tutorial 4 1. Show that if PQR is a triangle and is an affine transformation which maps P to Q , Q to R and R to P then ( i ) 3 = the identity ( ii ) is an isometry if and only if the triangle PQR is equilateral. 2. Which of the following maps of the plane to itself are affine transformations? Which are isometries? Classify those that are isometries. ( i ) ( x,y ) = (2 x 2 + 3 y + 3 , 4 y + 1) ( ii ) ( x,y ) = (- y + 4 ,x- 7) ( iii ) ( x,y ) = (2 x + 4 y- 7 ,x + 2 y + 1) ( iv ) ( x,y ) = ( y, 3 x + 4) 3. Let A = parenleftbigg u 1 v 1 u 2 v 2 parenrightbigg be an orthogonal matrix. Let u = parenleftbigg u 1 u 2 parenrightbigg , v = parenleftbigg v 1 v 2 parenrightbigg . ( i ) Show that u and v are perpendicular unit vectors. ( ii ) Show that A is, for some , either parenleftbigg cos - sin sin cos parenrightbigg or parenleftbigg cos sin sin...
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