College Algebra Exam Review 168

College Algebra Exam Review 168 - 3.3.36 , W has a...

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178 3. PRODUCTS OF GROUPS Proof. Parts (a) through (c) are left to the reader as exercises. See Exercise 3.4.6 . For part (d), we have W ± W ıı , by part (c). Suppose that v 2 V but v 62 W . Consider the quotient map ± W V ²! V=W . Since ±.v/ ¤ 0 , by Lemma 3.4.4 , there exists g 2 .V=W / ± such that g.±.v// ¤ 0 . Write ± ± .g/ D g ı ± . We have ± ± .g/ 2 W ı but h v;± ± .g/ i ¤ 0 . Thus v 62 W ıı . Since S ıı is a subspace of V containing S by parts (a) and (c), we have S ± span .S/ ± S ıı . Taking annihilators, and using part (b), we have S ııı ± span .S/ ı ± S ı . But S ı ± S ııı by part (c), so all these sub- spaces are equal. Taking annihilators once more gives S ıı D span .S/ ıı D span .S/ , where the final equality results from part (d). n With the aid of annihilators, we can describe the dual space of sub- spaces and quotients. Proposition 3.4.7. Let W be a subpace of a finite dimensional vector space V . The restriction map f 7! f j W is a surjective linear map from V ± onto W ± with kernel W ı . Consequently, W ± Š V ± =W ı . Proof. I leave it to the reader to check that f 7! f j W is linear and has kernel W ı . Let us check the surjectivity of this map. According to Proposition
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Unformatted text preview: 3.3.36 , W has a complement in V , so V D W M for some subspace M . We can use this direct sum decomposition to dene a surjective linear map from V to W with kernel M , namely .w C m/ D w , for w 2 W and m 2 M . Now for g 2 W , we have .g/ D g 2 V , and .g/.w/ D g..w// D g.w/ for w 2 W . Thus g is the restriction to W of .g/ . Finally, we have W V =W by the homomorphism theorem for vector spaces. n What about the dual space to V=W ? Let W V ! V=W denote the quotient map. For g 2 .V=W / , .g/ D g is an element of V that is zero on W , that is, an element of W . The proof of the following proposition is left to the reader. Proposition 3.4.8. The map g 7! .g/ D g is a linear isomorphism of .V=W / onto W . Proof. Exercise 3.4.8 . n...
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