A3 - V . Prove that the following are equivalent. 1) L-1...

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Math 235 Assignment 3 Due: Wednesday, Oct 6th 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) M (2 , 2) and P 3 . b) The vector space P = { p ( x ) P 2 | p (2) = 0 } and the vector space U of 2 × 2 diagonal matrices. 2. Suppose that { ~v 1 ,...,~v r } is a linearly independent set in a vector space V and that L : V W is a one-to-one linear map. Prove that { L ( ~v 1 ) ,...,L ( ~v r ) } is a linearly independent set in W . 3. Let L be a linear operator on an n dimensional vector space
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Unformatted text preview: V . Prove that the following are equivalent. 1) L-1 exists. 2) L is one-to-one. 3) Null( L ) = { ~ } 4) L is onto. 4. Let V and W be vector spaces with dim V = n and dim W = m , let L : V W be a linear mapping, and let A be the matrix of L with respect to bases B for V and C for W . a) Dene an explicit isomorphism from Range( L ) to Col( A ). Prove that your map is an isomorphism. b) Use a) to prove that rank( L ) = rank( A )....
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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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