College Algebra Exam Review 348

# College Algebra Exam Review 348 - D x ±².j Thus ±.².x...

This preview shows page 1. Sign up to view the full content.

358 8. MODULES for all x 1 ; : : : ; x n 2 M and all 2 S n . (c) A multilinear function ' W M n ! N is said to be alternating if '.x 1 ; : : : ; x n / D 0 whenever x i D x j for some i ¤ j . Lemma 8.3.3. The symmetric group acts S n on the set of multilinear func- tions from M n to N by the formula '.x 1 ; : : : ; x n / D '.x .1/ ; : : : ; x .n/ /: The set of symmetric (resp. skew–symmetric, alternating) multilinear func- tions is invariant under the action of S n . Proof. We leave it to the reader to check that ' is multilinear if ' is multilinear, and also that if ' is symmetric (resp. skew–symmetric, alter- nating), then ' satisfies the same condition. See Exercise 8.3.2 . To check that S n acts on ˚ n , we have to show that . /' D . '/ . Note that . '/.x 1 ; : : : ; x n / D . '/.x .1/ ; : : : ; x .n/ /: Now write y i D x .i/ for each i . Then also y .j /
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: D x ±².j/ . Thus, ±.²'/.x 1 ;:::;x n / D .²'/.y 1 ;:::;y n / D '.y ².1/ ;:::;y ².n/ / D '.x ±.².1// ;:::;x ±.².1// / D '.x ±².1/ :::;x ±².n/ / D .±²/'.x 1 ;:::;x n /: n Note that that a multilinear function is symmetric if, and only if ±' D ' for all ± 2 S n and skew–symmetric if, and only if, ±' D ³.±/' for all ± 2 S n . Lemma 8.3.4. An alternating multilinear function ' W M n ±! N is skew–symmetric. Proof. Fix any pair of indices i < j , and any elements x k 2 M for k different from i;j . Deﬁne ´.x;y/ W M 2 ±! N by ´.x;y/ D '.x 1 ;:::;x i ± 1 ;x;x i C 1 ;:::;x j ± 1 ;y;x j C 1 ;:::;x n /...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern