5. Let V be a vector space over C and let T: V → V be a linear transformation. (a) Give the definition of an eigenvector of T and the definition of an eigenvalue of T. 2 (b) If u, we V are eigenvectors for V with eigenvalues A and u respectively such that A μ₂ prove that u + w is not an eigenvector of T. (c) For all y C, prove that V₁ = {u V: T(u) = yu} is a subspace of V. (d) If y E C and V₁ = {0}, is y an eigenvalue of T? Justify your answer. (e) If the characteristic polynomial of T is Pr(t) = (t 1) (t− 3)(t + i)(t − i) find the minimum polynomial of T. Explain your answer carefully.
5. Let V be a vector space over C and let T: V → V be a linear transformation. (a) Give the definition of an eigenvector of T and the definition of an eigenvalue of T. 2 (b) If u, we V are eigenvectors for V with eigenvalues A and u respectively such that A μ₂ prove that u + w is not an eigenvector of T. (c) For all y C, prove that V₁ = {u V: T(u) = yu} is a subspace of V. (d) If y E C and V₁ = {0}, is y an eigenvalue of T? Justify your answer. (e) If the characteristic polynomial of T is Pr(t) = (t 1) (t− 3)(t + i)(t − i) find the minimum polynomial of T. Explain your answer carefully.
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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Transcribed Image Text:5. Let V be a vector space over C and let T: V → V be a linear transformation.
(a) Give the definition of an eigenvector of T and the definition of an eigenvalue of T.
2
(b) If u, we V are eigenvectors for V with eigenvalues A and u respectively such that A μ₂
prove that u + w is not an eigenvector of T.
(c) For all y C, prove that
V₁ = {u V: T(u) = yu}
is a subspace of V.
(d) If y C and V₁ =
= {0}, is y an eigenvalue of T? Justify your answer.
(e) If the characteristic polynomial of T is
Pr(t) = (t − 1)(t − 3)(t + i)(t − i)
find the minimum polynomial of T. Explain your answer carefully.
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