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

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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) P 3 and R 4 . Solution: We define L : P 3 R 4 by L ( a 3 x 3 + a 2 x 2 + a 1 x + a 0 ) = ( a 3 ,a 2 ,a 1 ,a 0 ). Linear: Let any two elements of P 3 be ~a = a 3 x 3 + a 2 x 2 + a 1 x + a 0 and ~ b = b 3 x 3 + b 2 x 2 + b 1 x + b 0 and let k R then L ( k~a + ~ b ) = L ( k ( a 3 x 3 + a 2 x 2 + a 1 x + a 0 ) + ( b 3 x 3 + b 2 x 2 + b 1 x + b 0 ) ) = L ( ( ka 3 + b 3 ) x 3 + ( ka 2 + b 2 ) x 2 + ( ka 1 + b 1 ) x + ka 0 + b 0 ) = ( ka 3 + b 3 ,ka 2 + b 2 ,ka 1 + b 1 ,ka 0 + b 0 ) = k ( a 3 ,a 2 ,a 1 ,a 0 ) + ( b 3 ,b 2 ,b 1 ,b 0 ) = kL ( ~a ) + L ( ~ b ) Therefore L is linear. One-to-one: Assume L ( ~a ) = L ( ~ b ). Then L ( a 3 x 3 + a 2 x 2 + a 1 x + a 0 ) = L ( b 3 x 3 + b 2 x 2 + b 1 x + b 0 ) ( a 3 ,a 2 ,a 1 ,a 0 ) = ( b 3 ,b 2 ,b 1 ,b 0 ) . This gives a 3 = b 3 , a 2 = b 2 , a 1 = b 1 , a 0 = b 0 hence ~a = ~ b so L is one-to-one. Onto: For any ( a 3 ,a 2 ,a 1 ,a 0 ) R 4 we have L ( a 3 x 3 + a 2 x 2 + a 1 x + a 0 ) = ( a 3 ,a 2 ,a 1 ,a 0 ) hence L is onto. Thus,
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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

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