7. Let V be a complex vector space and let T e L(V). Let r be the minimal polynomial of T and suppose the degree of r is m. (a) Show that U = span {I,T,T²,T°,..T™-1} is an m-dimensional subspace of L(V). (b) Show that T™m e U. (c) If T is invertible, is T-1 e U?

7. Let V be a complex vector space and let T e L(V). Let r be the minimal polynomial of T and suppose the degree of r is m. (a) Show that U = span {I,T,T²,T°,..T™-1} is an m-dimensional subspace of L(V). (b) Show that T™m e U. (c) If T is invertible, is T-1 e U?

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

7. Let V be a complex vector space and let T € L(V). Let r be the minimal polynomial of T and suppose the
degree of r is m.
(a) Show that
U = span {I,T, T²,7°,..T™-1}
is an m-dimensional subspace of L(V).
(b) Show that T e U.
(c) If T is invertible, is T-1 e U?

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