College Algebra Exam Review 150

# College Algebra Exam Review 150 - Proof Exercise 3.3.3 n...

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160 3. PRODUCTS OF GROUPS (d) Let V denote the complex vector space of C –valued continuous functions on the interval Œ0;1Ł and let g 2 V . The map f 7! R 1 0 f.t/g.t/dt is a linear transformation from V to C . Linear transformations are the homomorphisms in the theory of vector spaces; in fact, a linear transformation T W V ! W between vector spaces is a homomorphism of abelian groups that additionally satisﬁes T.˛v/ D ˛T.v/ for all ˛ 2 K and v 2 V . A linear isomorphism between vector spaces is a bijective linear transformation between them. Deﬁnition 3.3.6. A subspace of a vector space V is a (nonempty) subset that is a vector space with the operations inherited from V . As with groups, we have a criterion for a subset of a vector space to be a subspace, in terms of closure under the vector space operations: Proposition 3.3.7. For a nonempty subset of a vector space to be a sub- space, it sufﬁces that the subset be closed under addition and under scalar multiplication.
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Unformatted text preview: Proof. Exercise 3.3.3 . n Again as with groups, the kernel of a vector space homomorphism (linear transformation) is a subspace of the domain, and the range of a vector space homomorphism is a subspace of the codomain. Proposition 3.3.8. Let T W V ! W be a linear map between vector spaces. Then the range of T is a subspace of W and the kernel of T is a subspace of V . Proof. Exercise 3.3.5 . n Quotients and homomorphism theorems If V is a vector space over K and W is a subspace, then in particular W is a subgroup of the abelian group V , so we can form the quotient group V=W , whose elements are cosets v C W of W in V . The additive group operation in V=W is .x C W / C .y C W / D .x C y/ C W . Now attempt to deﬁne a multiplication by scalars on V=W in the obvious way: ˛.v C W / D .˛v C W / . We have to check that this this is well–deﬁned....
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