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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, w E V are eigenvectors for V with eigenvalues d and µ respectively such that A + µ, prove that u+ w is not an eigenvector of T. (c) For all y E C, prove that Vy = {u € V : T(u) = yu} is a subspace of V.

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, w E V are eigenvectors for V with eigenvalues d and µ respectively such that A + µ, prove that u+ w is not an eigenvector of T. (c) For all y E C, prove that Vy = {u € V : T(u) = yu} is a subspace of V.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 25EQ
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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, w E V are eigenvectors for V with eigenvalues d and µ respectively such that A + µ, prove that u+ w is not an eigenvector of T. (c) For all y E C, prove that Vy = {u € V : T(u) = yu} is a subspace of V. (d) If y E C and Vy = {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.
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, w E V are eigenvectors for V with eigenvalues d and µ respectively such that A + µ, prove that u+ w is not an eigenvector of T. (c) For all y E C, prove that Vy = {u € V : T(u) = yu} is a subspace of V. (d) If y E C and Vy = {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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