MATH 235 - (1) D : M (2 , 2) R dened by D ( A ) = det A ....

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Math 235 Assignment 1 Due: Wednesday, May 11th 1. Find a basis for the four fundamental subspace of A = 1 1 0 - 1 - 2 1 2 - 1 3 1 0 - 1 1 - 4 - 3 - 1 3 - 4 7 4 - 2 1 - 3 4 3 2. Let L : V W be a linear mapping. Prove that L ( ~ 0) = ~ 0 3. Let L be the set of all linear mappings L : V W with standard addition and scalar multiplication of linear mappings. Prove that (1) tL L for all L L and t R . (2) ( s + t ) L = sL + tL for all L L and s,t R . 4. Determine which of the following mappings are linear. If it is linear, prove it. If not, give a counterexample to show that it is not linear.
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Unformatted text preview: (1) D : M (2 , 2) R dened by D ( A ) = det A . (2) L : P 2 P 2 dened by L ( a + bx + cx 2 ) = ( a-b ) + ( b + c ) x 2 . (3) T : M (2 , 2) P 2 dened by T a b c d = a + ( ab ) x + ( abc ) x 2 . 5. Let L : P 2 M (2 , 2) be dened by L ( a + bx + cx 2 ) = -a-2 c 2 b-c-2 a + 2 c-2 b-c (a) Evaluate L (1 + 2 x + 3 x 2 ). (b) Find a polynomial p such that L ( p ) = 4 1-10 5 ....
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