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

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Unformatted text preview: Math 235 Assignment 2 Solutions 1. Consider the projection proj (1 ,- 2) : R 2 R 2 onto the line ~x = t 1- 2 , t R . Determine a geometrically natural basis B and determine the matrix of the transformation with respect to B . Solution: Pick ~v 1 = 1- 2 . We then pick ~v 2 = 2 1 so that it is orthogonal to ~v 1 . By geometrical arguments, a basis adapted to proj ~v 1 is B = { ~v 1 ,~v 2 } . To determine the matrix of proj (1 ,- 2) with respect to B , calculate the B coordinates of the images of the basis vectors: proj ~v 1 ( ~v 1 ) = ~v 1 = 1 ~v 1 + 0 ~v 2 proj ~v 1 ( ~v 2 ) = ~ 0 = 0 ~v 1 + 0 ~v 2 Hence, we get [proj ~v 1 ] B = 1 0 0 0 . 2. Find a basis for the range and kernel of the following linear mappings and verify the Rank-Nullity Theorem. (a) L : R 3 R 2 defined by L ( x 1 ,x 2 ,x 3 ) = ( x 1 + x 2 ,x 1 + x 2 + x 3 ). Solution: For any ~x R 3 , we have L ( x 1 ,x 2 ,x 3 ) = x 1 + x 2 x 1 + x 2 + x 3 = ( x 1 + x 2 ) 1 1 + x 3 1 Thus, a basis for Range( L ) is Span 1 1 , 1 . If ~x Ker( L ), then = L ( ~x ) = x 1 + x 2 x 1 + x 2 + x 3 Thus, x 1 + x 2 = 0 and x 1 + x 2 + x 3 = 0. This gives x 3 = 0 and x 1 =- x 2 . Therefore, ~x = x 1 x 2 x 3 = - x 2 x 2 = x 2 - 1 1 Thus, a basis for Ker( L ) is - 1 1 ....
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A2_soln - Math 235 Assignment 2 Solutions 1. Consider the...

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