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Unformatted text preview: 3.4. THE DUAL OF A VECTOR SPACE AND MATRICES 173 Hint: Show (a) H) (b) H) (c) H) (a). For (c) H) (a), show that condition (c) implies that V has a finite maximal linearly independent subset. This is slightly easier if you use Zorns lemma, but Zorns lemma is not required. 3.3.12. Show that the following conditions are equivalent for a vector space V : (a) V is infinitedimensional. (b) V has an infinite linearly independent subset. (c) For every n 2 N , V has a linearly independent subset with n elements. 3.3.13. Prove Corollary 3.3.37 directly by using Corollary 3.3.28 , as fol lows: Let f v 1 ;v 2 ;:::;v s g be a basis of N . Then there exist vectors v s C 1 ;:::;v n such that f v 1 ;v 2 ;:::;v s ;v s C 1 ;:::;v n g is a basis of V . Let M D span . f v s C 1 ;:::;v n g / . Show that V D M N . 3.4. The dual of a vector space and matrices Let V and W be vector spaces over a field K . We observe that the set Hom K .V;W / of linear maps from V and W also has the structure of a...
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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
 EVERAGE
 Algebra, Matrices, Vector Space

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