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College Algebra Exam Review 347

# College Algebra Exam Review 347 - j ± 1;x;x j C 1;x n is...

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8.3. MULTILINEAR MAPS AND DETERMINANTS 357 8.2.4. Complete the proof of the Correspondence Theorem, Proposition 8.2.7 . 8.2.5. Prove Proposition 8.2.8 . 8.2.6. Prove the Factorization Theorem, Proposition 8.2.9 . 8.2.7. Prove the Diamond Isomorphism Theorem, Proposition 8.2.10 . 8.2.8. Let R be a ring with identity element. Let M be a finitely generated R –module. Show that there is a free R module F and a submodule K F such that M Š F=K as R –modules. 8.3. Multilinear maps and determinants Let R be a ring with multiplicative identity element. All R –modules will be assumed to be unital. Definition 8.3.1. Suppose that M 1 ; M 2 ; : : : ; M n and N are modules over R . A function ' W M 1 M n ! N is multilinear (or R –multilinear) if for each j and for fixed elements x i 2 M i ( i ¤ j ), the map x 7! '.x 1 ; : : : ; x
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Unformatted text preview: j ± 1 ;x;x j C 1 ;:::;x n / is an R –module homomorphism. It is easy to check that the set of all multilinear maps ' W M 1 ² ³³³ ² M n ´! N is an abelian group under addition; see Exercise 8.3.1 . We will be interested in the special case that all the M i are equal. In this case we can consider the behavior of ' under permutation of the variables. Deﬁnition 8.3.2. (a) A multilinear function ' W M n ´! N is said to be symmetric if '.x ±.1/ ;:::;x ±.n/ / D '.x 1 ;:::;x n / for all x 1 ;:::;x n 2 M and all ± 2 S n . (b) A multilinear function ' W M n ´! N is said to be skew– symmetric if '.x ±.1/ ;:::;x ±.n/ / D ².±/'.x 1 ;:::;x n /...
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