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Assignment Math 3
Due: Wednesday, May 26th
1. For each of the following pairs of vector spaces, dene an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a) P3 and R4 . b) The vector space P = {p(x) P3 | p(1) = 0} and the vector space U of 2 2 upper triangular matrices. 2. Suppose that {v1 , . . . , vr } is a linearly independent set in a vector space V and that L : V W is a one-to-one linear map. Prove {L(v1 that ), . . . , L(vr )} is a linearly independent set in W . 3. Let L be a linear operator on an n dimensional vector space V . Prove that the following are equivalent. 1) L1 exists. 2) L is one-to-one. 3) null(L) = {0} 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 . Dene T : S M (n, n) by T (L) = [L]B . a) Prove that T is an isomorphism. b) Use a) to nd a basis for S .