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 homomorphism 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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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Multiplication, Scalar, Vector Space, 2 K, t W, surjective linear map

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