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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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- Fall '08