assign5 - Math 136 Assignment 5 Due: Wednesday, Feb 24th 1....

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Unformatted text preview: Math 136 Assignment 5 Due: Wednesday, Feb 24th 1. Let A = a) 2A - B 1 2 -1 1 0 0 5 -1 2 ,B= ,C= . Determine the following 3 -2 -1 0 2 -3 1 -1 2 b) A(B T + C T ) c) BAT + CAT 2. Prove that if x M (3, 2) and a, b R are scalars, then (a + b)x = ax + bx. 3. Determine which of the following mappings are linear. Find the standard matrix of each linear mapping. a) f (x1 , x2 , x3 ) = (x1 + x2 + 1, x3 , 0). b) f (x1 , x2 ) = (0, x1 + 2x2 , x2 ). c) proj(-2,1) . 4. Determine the standard matrix of a reflection in R2 in the line x1 - 5x2 = 0. 5. Let L and M be linear mappings from Rn to Rm , and let k R. a) Prove that L + M and kL are linear mappings. b) Prove that [kL + M ] = k[L] + [M ]. 6. Suppose that S and T are linear mappings with matrices 4 -3 1 4 0 2 3 1 [T ] = . [S] = 5 -3 -2 1 3 0 -2 0 a) Determine the domain and codomain of each mapping. b) Determine the standard matrices that represent S T and T S. 1 2 1 1 0 3 7. Let L be the linear mapping with matrix [L] = 2 1 -1. 0 2 2 a) Is (5, 5, 4, 4) in the range of L? If so, find v such that L(v) = (5, 5, 4, 4). b) Is (1, -3, 2, 1) in the range of L? If so, find v such that L(v) = (1, -3, 2, 1). c) Determine a spanning set for the nullspace of L. 8. Determine the matrix of a linear mapping L : R2 R2 whose nullspace is span{(2, 1)} and whose range is span{(2, -1)}. 1 ...
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