College Algebra Exam Review 162

College Algebra Exam Review 162 - 2.7.18 , or by adapting...

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172 3. PRODUCTS OF GROUPS Warning: Complements of a subspace are never unique. For example, both f .0;0;c/ W c 2 R g and f .0;c;c/ W c 2 R g are complements of f .a;b;0/ W a;b 2 R g in R 3 . Exercises 3.3 3.3.1. Show that the intersection of an arbitrary family of linear subspaces of a vector space is a linear subspace. 3.3.2. Let S be a subset of a vector space. Show that span .S/ D span . span .S// . Show that span .S/ is the unique smallest linear subspace of V containing S as a subset, and that it is the intersection of all linear subspaces of V that contain S as a subset. 3.3.3. Prove Proposition 3.3.7 . 3.3.4. Show that any composition of linear transformations is linear. Show that the inverse of a linear isomorphism is linear. 3.3.5. Let T W V ! W be a linear map between vector spaces. Show that the range of T is a subspace of W and the kernel of T is a subspace of V . 3.3.6. Prove Proposition 3.3.11 . 3.3.7. Give another proof of Proposition 3.3.12 by adapting the proof of Proposition 2.7.13 rather than by using that proposition. 3.3.8. Prove Proposition 3.3.14 by using Proposition
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Unformatted text preview: 2.7.18 , or by adapting the proof of that proposition. 3.3.9. Let A and B be nitedimensional subspaces of a not necessarily nitedimensional vector space V . Show that A C B is nitedimensional and that dim .A C B/ C dim .A \ B/ D dim .A/ C dim .B/ . 3.3.10. Let V be a vector space over K (a) Let A and B be matrices over K of size n by s and s by t re-spectively. Show that for v 1 ;:::;v n 2 K n , v 1 ;:::;v n .AB/ D .v 1 ;:::;v n A/B: (b) Show that if f v 1 ;:::;v n g is linearly independent subset of V , and v 1 ;:::;v n A D , then A D . 3.3.11. Show that the following conditions are equivalent for a vector space V : (a) V is nite dimensional. (b) Every linearly independent subset of V is nite. (c) V does not admit an innite, strictly increasing sequence of lin-early independent subsets Y 1 Y 2 ::: ....
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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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