Let V be a vector space of dimension n > 1 and suppose TE L(V). Here T is a fixed map which will not change for the rest of the exam. Assume further that T is not the zero map. 1. Consider the mapping Pr: Pa (C) L(V) given by Pr(p) = p(T). Show that Pr is a linear map. 2. Show that the dimension of null Pr is at least 1. 3. Show that if p E null Pr and A is an eigenvalue of T' then p(A) = 0. Since T + 0 as a map, any non-zero polynomial in null Pr must have positive degree. Let k be the smallest positive integer so that there ia a polynomial p in null Pr of degree k. Let Pm be the unique polynomial of degree k in null Pr that has leading coefficient 1. (If there were two such polynomials then the difference would still be in null Pr but would have degree k-1.) We will call pm the minimal polynomial of T. 4. Suppose Pm(A) = 0 for some A e C. Prove that A is an eigenvalue of T.

Let V be a vector space of dimension n > 1 and suppose TE L(V). Here T is a fixed map which will not change for the rest of the exam. Assume further that T is not the zero map. 1. Consider the mapping Pr: Pa (C) L(V) given by Pr(p) = p(T). Show that Pr is a linear map. 2. Show that the dimension of null Pr is at least 1. 3. Show that if p E null Pr and A is an eigenvalue of T' then p(A) = 0. Since T + 0 as a map, any non-zero polynomial in null Pr must have positive degree. Let k be the smallest positive integer so that there ia a polynomial p in null Pr of degree k. Let Pm be the unique polynomial of degree k in null Pr that has leading coefficient 1. (If there were two such polynomials then the difference would still be in null Pr but would have degree k-1.) We will call pm the minimal polynomial of T. 4. Suppose Pm(A) = 0 for some A e C. Prove that A is an eigenvalue of T.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let V be a vector space of dimension n > 1 and suppose TE L(V). Here
T is a fixed map which will not change for the rest of the exam. Assume
further thatT is not the zero map.
1. Consider the mapping Pr: Pna(C) L(V) given by Pr(p) = p(T).
Show that Pr is a linear map.
2. Show that the dimension of null Pr is at least 1.
3. Show that if pe null Pr and X is an eigenvalue of T then p(X) = 0.
Since T # 0 as a map, any non-zero polynomial in null Pr must have
positive degree. Let k be the smallest positive integer so that there ia a
polynomial p in null Pr of degree k. Let Pm be the unique polynomial of
degree k in null Pr that has leading coefficient 1. (If there were two such
polynomials then the difference would still be in null Pr but would have
degree k- 1.) We will call pm the minimal polynomial of T.
4. Suppose pm(A) = 0 for some AE C. Prove that A is an eigenvalue of
т.
%3D
%3D

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