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Unformatted text preview: Math 235 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 a b c d = ax 3 + bx 2 + cx + d . To prove that it is an isomorphism, we must prove that it is linear, onetoone and onto. Linear: Let any two elements of M (2 , 2) be ~a = a 1 b 1 c 1 d 1 and ~ b = a 2 b 2 c 2 d 2 and let k ∈ R then L ( k~a + ~ b ) = L ( k a 1 b 1 c 1 d 1 + a 2 b 2 c 2 d 2 ) = L ( ka 1 + a 2 kb 1 + b 2 kc 1 + c 2 kd 1 + d 2 ) = ( ( 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. Onetoone: Assume L ( ~a ) = L ( ~ b ). Then L a 1 b 1 c 1 d 1 = L a 2 b 2 c 2 d 2 ⇒ 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 ....
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This note was uploaded on 04/18/2010 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.
 Spring '08
 CELMIN
 Math, Vector Space

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