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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 halftranslated German words. The German Eigenvektor and Eigenwert mean “characteristic vector” 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.
 Fall '08
 EVERAGE
 Algebra, Factors

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