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(a) Let T be a linear operator on a vector space V, and let y be a cycle of generalized eigenvectors that corresponds to the eigenvalue A. Prove that Span(7) is a T-invariant subspace of V.

(a) Let T be a linear operator on a vector space V, and let y be a cycle of generalized eigenvectors that corresponds to the eigenvalue A. Prove that Span(7) is a T-invariant subspace of V.

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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Supply a proof of each of the statements below. (a) Let T be a linear operator on a vector space V, and let y be a cycle of generalized eigenvectors that corresponds to the eigenvalue A. Prove that Span(7) is a T-invariant subspace of V. (b) Let T be a linear operator on a vector space V, let v be a nonzero element of V, and let W be the T-cyclic subspace of V generated by v. For any w e V, prove that w e W if and only if there exists a polynomial g(t) such that w = g(T)(v).
Transcribed Image Text:Supply a proof of each of the statements below. (a) Let T be a linear operator on a vector space V, and let y be a cycle of generalized eigenvectors that corresponds to the eigenvalue A. Prove that Span(7) is a T-invariant subspace of V. (b) Let T be a linear operator on a vector space V, let v be a nonzero element of V, and let W be the T-cyclic subspace of V generated by v. For any w e V, prove that w e W if and only if there exists a polynomial g(t) such that w = g(T)(v).
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