College Algebra Exam Review 151

College Algebra Exam Review 151 - ±.˛v D ˛±.v for v 2 V...

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3.3. VECTOR SPACES 161 But this follows from the closure of W under scalar multiplication; namely, if v C W D v 0 C W and, then ˛v ± ˛v 0 D ˛.v ± v 0 / 2 ˛W ² W . Thus ˛v C W D ˛v 0 C W , and the scalar multiplication on V=W is well-defined. Theorem 3.3.9. If W is subspace of a vector space V over K , then V=W has the structure of a vector space, and the quotient map ± W v 7! v C W is a surjective linear map from V to V=W with kernel equal to W . Proof. We know that V=W has the structure of an abelian group, and that, moreover, there is a well-defined product K ³ V=W ±! V=W given by ˛.v C W / D ˛v C W . It is straighforward to check the remaining vector space axioms. Let us indclude one verification for the sake of illustration. For ˛ 2 K and v 1 ;v 2 2 V , ˛..v 1 C W / C .v 2 C W // D ˛..v 1 C v 2 / C W // D ˛.v 1 C v 2 / C W D .˛v 1 C ˛v 2 / C W D .˛v 1 C W / C .˛v 2 C W / D ˛.v 1 C W / C ˛.v 2 C W / Finally, the quotient map ± is already known to be a group homomorophism. To check that it is linear, we only need to verify that
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Unformatted text preview: ±.˛v/ D ˛±.v/ for v 2 V and ˛ 2 K . But this is immediate from the definition of the product, ˛v C W D ˛.v C W / . n V=W is called the quotient vector space and v 7! v C W the quotient map or quotient homomorphism . We have a homomorphism theorem for vector spaces that is analogous to, and in fact follows from, the homomor-phism theorem for groups. Theorem 3.3.10. (Homomorphism theorem for vector spaces). Let T W V ±! V be a surjective linear map of vector spaces with kernel N . Let ± W V ±! V=N be the quotient map. There is linear isomorphism Q T W V=N ±! V satisfying Q T ı ± D T . (See the following diagram.) V T q q q q q V ± q q q q q ± ± ± ± ± ± q q q q q q q ∼ = Q T V=N...
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