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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) perp (2 , 1 , 2) Solution: Pick ~v 1 = (2 , 1 , 2). We want to pick two more vectors that are orthogonal to (2 , 1 , 2). We pick ~v 2 = (1 , 2 , 0) and ~v 3 = (0 , 2 , 1). By geometrical arguments, a basis adapted to perp (2 , 1 , 2) is B = { ~v 1 ,~v 2 ,~v 3 } . To determine the matrix of perp (2 , 1 , 2) with respect to B , calculate the B coordinates of the images of the basis vectors: perp (2 , 1 , 2) ( ~v 1 ) = ~ 0 = 0 ~v 1 + 0 ~v 2 + 0 ~v 3 perp (2 , 1 , 2) ( ~v 2 ) = ~v 2 = 0 ~v 1 + 1 ~v 2 + 0 ~v 3 perp (2 , 1 , 2) ( ~v 3 ) = ~v 3 = 0 ~v 1 + ~v 2 + 1 ~v 3 Hence, we get [perp (2 , 1 , 2) ] B = 0 0 0 0 1 0 0 0 1 . b) refl (1 , 2 , 2) Solution: Consider the vector ~v 1 = (1 , 2 , 2), which is the normal of the plane of reflection, and the vectors ~v 2 = (2 , 1 ,...
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 Spring '10
 WILKIE
 Transformations

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