College Algebra Exam Review 153

College Algebra Exam Review 153 - Q T .˛.v C N// D Q T...

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3.3. VECTOR SPACES 163 Proposition 3.3.13. (Factorization Theorem for Vector Spaces) Let V and V be vector spaces over a field K , and let T W V ±! V be a surjec- tive linear map with kernel M . Let N ² M be a vector subspace and let ± W V ±! V=N denote the quotient map. The there is a surjective homomorphism Q T W V=N ±! V such that Q T ı ± D T . (See the following diagram.) The kernel of Q T is M=N ² V=N . V T q q q q q V ± q q q q q ± ± ± ± ± ± q q q q q q q Q T V=N Proof. By Proposition 2.7.14 , Q T W v C N 7! T.v/ defines a group homo- morphism from V=N onto V with kernel M=N . We only have to check that this map respects multiplication by elements of K . This follows from the computation:
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Unformatted text preview: Q T .˛.v C N// D Q T .˛v C N/ D T.˛v/ D ˛T.v/ D ˛ Q T .v C N/: n Proposition 3.3.14. (Diamond Isomorphism Theorem for Vector Spaces) Let A and N be subspaces of a vector space V . Let ± denote the quotient map ± W V ! V=N . Then ± ± 1 .±.A// D A C N is a subspace of V containing both A and N . Furthermore, .A C N/=N Š ±.A/ Š A=.A \ N/ . Proof. Exercise 3.3.8 . n Bases and dimension We now consider span, linear independence, bases and dimension for abstract vector spaces....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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