assign5 - Math 136 Assignment 5 Due: Wednesday, Feb 15th 1....

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Unformatted text preview: Math 136 Assignment 5 Due: Wednesday, Feb 15th 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. Find t1 , t2 , t3 R such that t1 1 1 2 0 0 2 6 3 + t2 + t3 = 0 1 0 2 -1 1 3 3 3. Determine which of the following mappings are linear. Find the standard matrix, a basis for the kernel, and a basis for the range of each linear mapping. (a) f (x1 , x2 , x3 ) = (x1 + x2 + 1, x3 , 0). (b) f (x1 , x2 ) = (0, x1 + 2x2 , x2 ). (c) proja where a = -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. Let L : Rn Rm be a linear mapping. Prove that 0 Ker(L). 7. Let L : Rn Rm be a linear mapping such that Ker(L) = {0} and Range(L) = Rm . Prove that m = n. 8. Determine the matrix of a linear mapping L : R2 R2 whose kernel is Span{(2, 1)} and whose range is Span{(2, -1)}. 1 ...
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This note was uploaded on 02/22/2012 for the course CS cs136 taught by Professor Cormack during the Winter '10 term at Waterloo.

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