A2_soln - Math 235 Assignment 2 Solutions 1 Determine the...

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Unformatted text preview: Math 235 Assignment 2 Solutions 1. Determine the matrix of the linear operator L : R 3 → R 3 with respect to the basis B and determine [ L ( ~x )] B where B = { ~v 1 ,~v 2 ,~v 3 } , L ( ~v 1 ) = ~v 1 +2 ~v 2- ~v 3 , L ( ~v 2 ) = 2 ~v 1- 2 ~v 2 + ~v 3 , L ( ~v 3 ) = ~v 2 + 3 ~v 3 and ( ~x ) B = (1 , 2 ,- 1). Solution: To determine the matrix of L with respect to B , we need the B-coordinates of the images of the basis vectors. We have [ L ( ~v 1 )] B = (1 , 2 ,- 1) , [ L ( ~v 2 )] B = (2 ,- 2 , 1) , [ L ( ~v 3 )] B = (0 , 1 , 3) . Hence, the matrix of L with respect to B is [ L ] B = [ L ( ~v 1 )] B [ L ( ~v 2 )] B [ L ( ~v 3 )] B = 1 2 2- 2 1- 1 1 3 . Thus [ L ( ~x )] B = [ L ] B [ ~x ] B = 1 2 2- 2 1- 1 1 3 1 2- 1 = 5- 3- 2 . 2. 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 ,...
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A2_soln - Math 235 Assignment 2 Solutions 1 Determine the...

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