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# A3 - 3 Let L be a linear operator on an n dimensional...

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Math 235 Assignment 3 Due: Wednesday, May 26th 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 . b) The vector space P = { p ( x ) P 3 | p (1) = 0 } and the vector space U of 2 × 2 upper triangular 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
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Unformatted text preview: . 3. Let L be a linear operator on an n dimensional vector space 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 be an n dimensional vector space with basis B and let S be the vector space of all linear operators L : V → V . Deﬁne T : S → M ( n,n ) by T ( L ) = [ L ] B . a) Prove that T is an isomorphism. b) Use a) to ﬁnd a basis for S ....
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