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Consider T E L(V). (a) Suppose U is a subspace of V invariant under T. Prove that U is also invariant under p(T) for every polynomial p E P(IF). (b) Suppose v is an eigenvector of T with eigenvalue A. Suppose pE P(F). (b1) Show that p(T)v = p(1)v. (b2) Is v an eigenvector for p(T)? If so, for what eigenvalue?

Consider T E L(V). (a) Suppose U is a subspace of V invariant under T. Prove that U is also invariant under p(T) for every polynomial p E P(IF). (b) Suppose v is an eigenvector of T with eigenvalue A. Suppose pE P(F). (b1) Show that p(T)v = p(1)v. (b2) Is v an eigenvector for p(T)? If so, for what eigenvalue?

Advanced Engineering Mathematics
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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Consider T E L(V). (a) Suppose U is a subspace of V invariant under T. Prove that U is also invariant under p(T) for every polynomial p E P(IF). (b) Suppose v is an eigenvector of T with eigenvalue A. Suppose pE P(F). (b1) Show that p(T)v = p(1)v. (b2) Is v an eigenvector for p(T)? If so, for what eigenvalue?
Transcribed Image Text:Consider T E L(V). (a) Suppose U is a subspace of V invariant under T. Prove that U is also invariant under p(T) for every polynomial p E P(IF). (b) Suppose v is an eigenvector of T with eigenvalue A. Suppose pE P(F). (b1) Show that p(T)v = p(1)v. (b2) Is v an eigenvector for p(T)? If so, for what eigenvalue?
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