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4. Let V be a vector space over C and let T : V → V be a linear transformation. (a) State the definition of an eigenvector of T. (b) If 0 is an eigenvalue of T, show that T is not invertible. (c) Find the eigenvectors and eigenvalues of the matrix A given below. -1 0 0 A = 10 3 1 6 0 2

4. Let V be a vector space over C and let T : V → V be a linear transformation. (a) State the definition of an eigenvector of T. (b) If 0 is an eigenvalue of T, show that T is not invertible. (c) Find the eigenvectors and eigenvalues of the matrix A given below. -1 0 0 A = 10 3 1 6 0 2

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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4. Let V be a vector space over C and let T: V → V be a linear transformation. (a) State the definition of an eigenvector of T. (b) If 0 is an eigenvalue of T, show that T is not invertible. (c) Find the eigenvectors and eigenvalues of the matrix A given below. -1 0 0 A = 10 3 1 6 0 2 (d) (i) State the definition of a diagonalisable matrix. (ii) Using theorems from class to justify your answer, explain why or why not the matrix B below is diagonalisable. 3 -5 3 2 B = -5 0 -10 2 -10 (e) Let V = Mat2x2(C). A linear transformation T: V → V is defined by a b T: -a с d b Find the eigenvalues and eigenvectors of T. (Hint: Work from the definition to find eigenvalues.) 2.
Transcribed Image Text:4. Let V be a vector space over C and let T: V → V be a linear transformation. (a) State the definition of an eigenvector of T. (b) If 0 is an eigenvalue of T, show that T is not invertible. (c) Find the eigenvectors and eigenvalues of the matrix A given below. -1 0 0 A = 10 3 1 6 0 2 (d) (i) State the definition of a diagonalisable matrix. (ii) Using theorems from class to justify your answer, explain why or why not the matrix B below is diagonalisable. 3 -5 3 2 B = -5 0 -10 2 -10 (e) Let V = Mat2x2(C). A linear transformation T: V → V is defined by a b T: -a с d b Find the eigenvalues and eigenvectors of T. (Hint: Work from the definition to find eigenvalues.) 2.
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