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

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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) R 4 and M (2 , 2). Solution: We define L : R 4 M (2 , 2) by L ( a 1 ,a 2 ,a 3 ,a 4 ) = a 1 a 2 a 3 a 4 . 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 R 4 be ~a = ( a 1 ,a 2 ,a 3 ,a 4 ) and ~ b = ( b 1 ,b 2 ,b 3 ,b 4 ) and let k R then L ( k~a + ~ b ) = L ( k (( a 1 ,a 2 ,a 3 ,a 4 ) + ( b 1 ,b 2 ,b 3 ,b 4 ) ) = L ( ka 1 + b 1 ,ka 2 b 2 ,ka 3 + b 3 ,ka 4 + b 4 ) = ka 1 + b 1 ka 2 + b 2 ka 3 + b 3 ka 4 + b 4 = k a 1 a 2 a 3 a 4 + b 1 b 2 b 3 b 4 = kL ( ~a ) + L ( ~ b ) Therefore L is linear. One-to-one: Assume L ( ~a ) = L ( ~ b ). Then L ( a 1 ,a 2 ,a 3 ,a 4 ) = L (( b 1 ,b 2 ,b 3 ,b 4 ) a 1 a 2 a 3 a 4 = b 1 b 2 b 3 b 4 . This gives a 1 = b 1 , a 2 = b 2 , a 3 = b 3 , a 4 = b 4 hence ~a = ~ b so L is one-to-one. Onto: For any a 1 a 2 a 3 a 4 M (2 , 2) we have L ( a 1 ,a 2 ,a 3 ,a 4 ) = a 1 a 2 a 3 a 4 hence L is onto....
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This note was uploaded on 10/12/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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

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