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College Algebra Exam Review 167

College Algebra Exam Review 167 - 3.4 THE DUAL OF A VECTOR...

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3.4. THE DUAL OF A VECTOR SPACE AND MATRICES 177 Lemma 3.4.4. Let V be a finite dimensional vector space over a field K . For each non-zero v 2 V , there is a linear functional f 2 V such that f .v/ ¤ 0 . Proof. We know that any linearly independent subset of V is contained in a basis. If v is a non-zero vector in V , then f v g is linearly independent. Therefore, there is a basis B of V with v 2 B . Let f be any function from B into K with f .v/ ¤ 0 . By Proposition 3.3.32 , f extends to a linear functional on V . n Theorem 3.4.5. If V is a finite dimensional vector space, then W V ! V is a linear isomorphism. Proof. We already know that is linear. If v is a non-zero vector in V , then there is an f 2 V such that f .v/ ¤ 0 , by Lemma 3.4.4 . Thus .v/.f / D f .v/ ¤ 0 , and .v/ ¤ 0 . Thus is injective. Applying Proposition 3.4.2 (c) twice, we have dim .V / D dim .V / D dim .V / . Therefore any injective linear map from V to V is necessarily surjective, by Proposition
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