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Unformatted text preview: Math 235 Assignment 3 Solutions 1. Prove that any plane through the origin in R 3 is isomorphic to R 2 . Solution: Since a plane through the origin in R 3 is a two-dimensional vector space, it is isomorphic to R 2 . 2. 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 2 and P 1 . Solution: We define L : R 2 → P 1 by L ( a,b ) = a + bx . To prove that it is an isomorphism, we must prove that it is linear, one-to-one and onto. Linear: Let ~x = a 1 b 1 ,~ y = a 2 b 2 ∈ R 2 and let s,t ∈ R then L ( s~x + t~ y ) = L ( s a 1 b 1 + t a 2 b 2 ) = L ( sa 1 + ta 2 sb 1 + tb 2 ) = ( ( sa 1 + ta 2 ) + ( sb 1 + tb 2 ) x ) = s ( a 1 + b 1 x ) + t ( a 2 + b 2 x ) = sL ( ~x ) + tL ( ~ y ) Therefore L is linear. One-to-one: Assume L ( ~x ) = L ( ~ y ). Then L a 1 b 1 = L a 2 b 2 ⇒ a 1 + b 1 x = a 2 + b 2 x This gives a 1 = a 2 and b 1 = b 2 , hence ~x = ~ y so L is one-to-one. Onto: For any a + bx ∈ P 3 we have L a b = a + bx hence L is onto....
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This note was uploaded on 09/30/2011 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.
- Spring '08
- Vector Space