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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## This note was uploaded on 11/28/2011 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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practiceprob3_soln - Math 235 Assignment 3 Practice...

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