practiceprob3_soln

practiceprob3_soln - Math 235 Assignment 3 Practice...

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Unformatted text preview: Math 235 Assignment 3 Practice Problems 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 + a 1 x + a 2 x 2 + a 3 x 3 ) = a a 1 a 2 a 3 . Linear: Let any two elements of P 3 be p ( x ) = a + a 1 x + a 2 x 2 + a 3 x 3 and q = b + b 1 x + b 2 x 2 + b 3 x 3 and let s,t R . Then L ( sp + tq ) = L ( s ( a + a 1 x + a 2 x 2 + a 3 x 3 ) + t ( b + b 1 x + b 2 x 2 + b 3 x 3 ) ) = L ( sa + tb + ( sa 1 + tb 1 ) x + ( sa 2 + tb 2 ) x 2 + ( sa 3 + tb 3 ) x 3 ) = sa + tb sa 1 + tb 1 sa 2 + tb 2 sa 3 + tb 3 = s a a 1 a 2 a 3 + t b b 1 b 2 b 3 = sL ( p ) + tL ( q ) Therefore L is linear. One-to-one: Assume L ( p ) = L ( q ). Then L ( a + a 1 x + a 2 x 2 + a 3 x 3 ) = L ( b + b 1 x + b 2 x 2 + b 3 x 3 ) a a 1 a 2 a 3 = b b 1 b 2 b 3 This gives a 3 = b 3 , a 2 = b 2 , a 1 = b 1 , a = b hence p = q so L is one-to-one....
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practiceprob3_soln - Math 235 Assignment 3 Practice...

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