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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 Fall '08
 EVERAGE
 Algebra, Transformations, Vector Space, 2 K, Quotients

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