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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.
- Fall '08