College Algebra Exam Review 394

# College Algebra Exam Review 394 - mial) of T 2 End K .V /...

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404 8. MODULES Proof. This is immediate, since ± T .x/ is the largest invariant factor of T , and ² T .x/ is the product of all of the invariant factors. n Let us make a few more remarks about the relation between the mini- mal polynomial and the characteristic polynomial. All of the invariant fac- tors of T divide the minimal polynomial ± T .x/ , and ² T .x/ is the product of all the invariant factors. It follows that ² T .x/ and ± T .x/ have the same irreducible factors, but with possibly different multiplicities. Since ³ 2 K is a root of a polynomial exactly when x ± ³ is an irreducible factor, we also have that ² T .x/ and ± T .x/ have the same roots, but with possibly different multiplicities. Finally, the characteristic polynomial and the min- imal polynomial coincide precisely if V is a cyclic KŒxŁ –module; i.e., the rational canonical form of T has only one block. Of course, statements analogous to Corollary 8.6.13 , and of these re- marks, hold for a matrix A 2 Mat n .K/ in place of the linear transformation T . The roots of the characteristic polynomial (or of the minimal polyno-
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Unformatted text preview: mial) of T 2 End K .V / have an important characterization. Deﬁnition 8.6.14. We say that an nonzero vector v 2 V is an eigenvector of T with eigenvalue ³ , if T v D ³v . Likewise, we say that a nonzero vector v 2 K n is an eigenvector of A 2 Mat n .K/ with eigenvalue ³ if Av D ³v . The words “eigenvector” and “eigenvalue” are half-translated German words. The German Eigenvektor and Eigenwert mean “characteristic vec-tor” and “characteristic value.” Proposition 8.6.15. Let T 2 End K .V / . An element ³ 2 K is a root of ² T .x/ if, and only if, T has an eigenvector in V with eigenvalue ³ . Proof. Exercise 8.6.7 n Exercises 8.6 8.6.1. Let h.x/ 2 KŒxŁ be a polynomial of one variable. Show that there is a polynomial g.x;y/ 2 KŒx;yŁ such that h.x/ ± h.y/ D .x ± y/g.x;y/ . 8.6.2. Consider f w 1 ;:::;w n g deﬁned by Equation 8.6.1 . Show that f w 1 ;:::;w n g is linearly independent over KŒxŁ ....
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## This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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