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# sol-a2 - University of Toronto at Scarborough Department of...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences Linear Programming and Optimazation MATB61 Winter 2008 Solution set to Assignment 2 Addition: 1. a) Let = + + + 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 4 3 2 1 r r r r where r i R, i = 1, 2, 3, 4. Then = + + + + 0 0 0 0 3 1 2 3 2 4 3 1 r r r r r r r r Solve the equations: r 1 + r 2 + r 4 = 0 r 2 + r 3 = 0 r 2 = 0 r 1 + r 3 = 0 It follows that r 1 = r 2 = r 3 = r 4 = 0. Therefore the set is independent. b) Let r 1 (1 + x) + r 2 (x + x 2 ) + r 3 (x 2 + x 3 ) + r 4 x 3 = 0 where r i R, i = 1, 2, 3, 4. Rearrange the equation, r 1 +( r 1 + r 2 )x + ( r 2 + r 3 )x 2 + ( r 3 + r 4 )x 3 = 0. Since {1, x, x 2 , x 3 } is independent, we have r 1 = 0 r 1 + r 2 = 0 r 2 + r 3 = 0 r 3 + r 4 = 0 It follows that r 1 = r 2 = r 3 = r 4 = 0. Therefore the set is independent. 2. a) Since {p(x) | p(x)= p( – x)} P 2 , Let p(x) = ax 2 + bx + c. If p(x) {p(x) | p(x)= p( – x)}, we have ax 2 + bx + c = a(- x) 2 + b(- x) + c = ax 2 – bx + c.

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sol-a2 - University of Toronto at Scarborough Department of...

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