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(a) Suppose V is a complex vector space and T € L(V). Show that if U is a subspace of V and U is invariant under T, then there exists u E U such that u is an eigenvector of T. [Hint: Consider Tv-] (b) Does the statement of (a) hold if V is a real vector space? Explain. (c) Does the statement of (a) hold if V is a real vector space and dim U = 1? Explain.

(a) Suppose V is a complex vector space and T € L(V). Show that if U is a subspace of V and U is invariant under T, then there exists u E U such that u is an eigenvector of T. [Hint: Consider Tv-] (b) Does the statement of (a) hold if V is a real vector space? Explain. (c) Does the statement of (a) hold if V is a real vector space and dim U = 1? Explain.

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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5. (a) Suppose V is a complex vector space and T€ L(V). Show that if U is a subspace of V and U is invariant under T, then there exists u e U such that u is an eigenvector of T. [Hint: Consider T|y] (b) Does the statement of (a) hold if V is a real vector space? Explain. (c) Does the statement of (a) hold if V is a real vector space and dim U = 1? Explain. (d) Let V be a complex vector space and S,T € L(V). Suppose ST = TS and let A be an eigenvalue of S. i. Show that E(A, S) is invariant under T. ii. What can you say about the eigenvectors of T? iii. Show that G(A, S) is invariant under T.
Transcribed Image Text:5. (a) Suppose V is a complex vector space and T€ L(V). Show that if U is a subspace of V and U is invariant under T, then there exists u e U such that u is an eigenvector of T. [Hint: Consider T|y] (b) Does the statement of (a) hold if V is a real vector space? Explain. (c) Does the statement of (a) hold if V is a real vector space and dim U = 1? Explain. (d) Let V be a complex vector space and S,T € L(V). Suppose ST = TS and let A be an eigenvalue of S. i. Show that E(A, S) is invariant under T. ii. What can you say about the eigenvectors of T? iii. Show that G(A, S) is invariant under T.
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