College Algebra Exam Review 349

College Algebra Exam Review 349 - A is skew–symmetric but...

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8.3. MULTILINEAR MAPS AND DETERMINANTS 359 By hypothesis, ± is R –bilinear and alternating: ±.x;x/ D 0 for all x 2 M . Therefore, 0 D ±.x C y;x C y/ D ±.x;x/ C ±.x;y/ C ±.y;x/ C ±.y;y/ D ±.x;y/ C ±.y;x/: Thus ±.x;y/ D ± ±.y;x/ . This shows that '.x ±.1/ ;:::;x ±.n/ / D . ± 1/'.x 1 ;:::;x n / when ² is the transposition .i;j/ . That is, ².'/ D ± ' , when ² D .i;j/ . In general, a permutation ² can be written as a product of transposi- tions, ² D ³ 1 ³ 2 ²²² ³ ` . Then ²' D ³ 1 2 . ²²² ³ ` .'/ ²²² // D . ± 1/ ` ' D ´.²/'; where we have used that S n acts on the set of multilinear functions and that ´ is a homomorphism from S n to 1 g . n Lemma 8.3.5. Let ' W M n ±! N be a multilinear function. Then S.'/ D P ± 2 S n ²' is a symmetric multilinear functional and A.'/ D P ± 2 S n ´.²/²' is an alternating multilinear functional. Proof. For ³ 2 S n , we have ³S.'/ D X ± 2 S n ³²' D X ± 2 S n ²' D S.'/; since ² 7! ³² is a bijection of S n . A similar argument shows that
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Unformatted text preview: A.'/ is skew–symmetric, but we have to work a little harder to show that A.'/ is alternating. Let x 1 ;x 2 ;:::;x n 2 M , and suppose that x i D x j for some i < j . The symmetric group S n is the disjoint union of the alternating group A n and its left coset .i;j/A n , where A n denotes the group of even permuta-tions, and .i;j/ is the transposition that interchanges i and j , and leaves all other points fixed. Thus, A.'/.x 1 ;:::;x n / D X ± 2 S n ´.²/²'.x 1 ;:::;x n / D X ± 2 A n .²'.x 1 ;:::;x n / ± .i;j/²'.x 1 ;:::;x n // D X ± 2 A n .'.x ±.1/ ;:::;x ±.n/ / ± '.x .i;j/±.1/ ;:::;x .i;j/±.n/ // I claim that each summand in this sum is zero....
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