A3_soln - Math 235 Assignment 3 Solutions 1 For each of the...

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Assignment 3 Solutions 1. For each of the following pairs of vector spaces, define an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) M (2 , 2) and P 3 . Solution: We define L : M (2 , 2) P 3 by L ±² a b c d ³´ = ax 3 + bx 2 + cx + d . To prove that it is an isomorphism, we must prove that it is linear, one-to-one and onto. Linear: Let any two elements of M (2 , 2) be ~a = ² a 1 b 1 c 1 d 1 ³ and ~ b = ² a 2 b 2 c 2 d 2 ³ and let k R then L ( k~a + ~ b ) = L ( k ² a 1 b 1 c 1 d 1 ³ + ² a 2 b 2 c 2 d 2 ³ ) = L ( ² ka 1 + a 2 kb 1 + b 2 kc 1 + c 2 kd 1 + d 2 ³ ) = ( ( ka 1 + a 2 ) x 3 + ( kb 1 + b 2 ) x 2 + ( kc 1 + c 2 ) x + ( kd 1 + d ) ) = k ( a 1 x 3 + b 1 x 2 + c 1 x + d 1 ) + a 2 x 3 + b 2 x 2 + c 2 x + d 2 = kL ( ~a ) + L ( ~ b ) Therefore L is linear. One-to-one: Assume L ( ~a ) = L ( ~ b ). Then L ±² a 1 b 1 c 1 d 1 ³´ = L ±² a 2 b 2 c 2 d 2 ³´ a 1 x 3 + b 1 x 2 + c 1 x + d 1 = a 2 x 3 + b 2 x 2 + c 2 x + d 2 . This gives a 1 = a 2 , b 1 = b 2 , c 1 = c 2 , d 1 = d 2 hence ~a = ~ b so L is one-to-one. Onto: For any ax 3 + bx 2 + cx + d P 3 we have L ±² a b c d ³´ = ax 3 + bx 2 + cx + d hence L is onto. Thus,
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A3_soln - Math 235 Assignment 3 Solutions 1 For each of the...

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