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Unformatted text preview: Math 235 Assignment 2 Solutions 1. For each of the following linear transformations, determine a geometrically natural basis B and determine the matrix of the transformation with respect to B . a) The projection proj (3 , 2) : R 2 → R 2 onto the line ~x = t 3 2 , t ∈ R . Solution: Pick ~v 1 = 3 2 . We then pick ~v 2 = 2 3 so that it is orthogonal to ~v 1 . By geometrical arguments, a basis adapted to proj (3 , 2) is B = { ~v 1 ,~v 2 } . To determine the matrix of proj (3 , 2) with respect to B , calculate the B coordinates of the images of the basis vectors: proj (3 , 2) ( ~v 1 ) = ~v 1 = 1 ~v 1 + 0 ~v 2 proj (3 , 2) ( ~v 2 ) = ~ 0 = 0 ~v 1 + 0 ~v 2 Hence, we get [proj (3 , 2) ] B = 1 0 0 0 . b) The reflection refl ( 1 , 1 , 1) : R 3 → R 3 over the plane with normal vector ~n =  1 1 1 . Solution: Consider the vector ~v 1 =  1 1 1 , which is the normal of the plane of reflection, and the vectors ~v 2 = 2 1 1 and ~v 3 = 1 1...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Transformations

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