College Algebra Exam Review 152

# College Algebra Exam Review 152 - B is a vector subspace of...

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162 3. PRODUCTS OF GROUPS Proof. The homomorphism theorem for groups (Theorem 2.7.6 ) gives us an isomorphism of abelian groups Q T satisfying Q T ı ± D T . We have only to verify that Q T also respects multiplication by scalars. But this follows at once from the deﬁnitions: Q T .˛.x C N// D Q T .˛x C N/ D T.˛x/ D ˛T.x/ D ˛ Q T .x C N/ . n The next three propositions are analogues for vector spaces and linear transformations of results that we have established for groups and group homomorphisms in Section 2.7 . Each is proved by adapting the proof from the group situation. Some of the details are left to you. Proposition 3.3.11. (Correspondence theorem for vector spaces) Let T W V ! V be a surjective linear map, with kernel N . Then M 7! T ± 1 . M/ is a bijection between subspaces of V and subspaces of V containing N . Proof. According to Proposition 2.7.12 , B 7! T ± 1 . B/ is a bijection be- tween the subgroups of V and the subgroups of V containing N . I leave it as an exercise to verify that
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Unformatted text preview: B is a vector subspace of V if, and only if, T ± 1 . B/ is a vector subspace of V ; see Exercise 3.3.6 . n Proposition 3.3.12. Let T W V ! V be a surjective linear transformation with kernel N . Let M be a subspace of V and let M D T ± 1 . M/ . Then x C M 7! T.x/ C M deﬁnes a linear isomorphism of V=M to V = M . Equivalently, .V=N/=.M=N/ Š V=M; as vector spaces. Proof. By Proposition 2.7.13 , the map x C M 7! T.x/ C M is a group iso-morphism from V=M to V = M . But the map also respects multiplication by elements of K , as ˛.v C M/ D ˛v C M 7! T.˛v/ C M D ˛T.v/ C M D ˛.T.v/ C M/ We can identify V with V=N , by the homomorphism theorem for vector spaces, and this identiﬁcation carries the subspace M to the image of M in V=N , namely M=N . Therefore .V=N/=.M=N/ Š V = M Š V=M: n...
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