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1. For each of linear transformation T given below, do the following: (i) find all eigenvalues of T, (ii) find each eigenspace of T and its basis, (iii) determine the algebraic and geometric multiplicities of each eigenvalue of T, (iv) determine if T is diagonalizable. (a) T: R² → R² defined by T(a, b) = (−2a + 3b, —10a +9b). (b) T: P₁ → P₁ defined by T(ax + b) = (-6a+2b)x + (−6a + b). (c) T: P3 → P3 defined by T(ƒ(x)) = f'(x) + ƒ"(x). (d) T: M2x2 → M2x2 defined by T(A) = AT.

1. For each of linear transformation T given below, do the following: (i) find all eigenvalues of T, (ii) find each eigenspace of T and its basis, (iii) determine the algebraic and geometric multiplicities of each eigenvalue of T, (iv) determine if T is diagonalizable. (a) T: R² → R² defined by T(a, b) = (−2a + 3b, —10a +9b). (b) T: P₁ → P₁ defined by T(ax + b) = (-6a+2b)x + (−6a + b). (c) T: P3 → P3 defined by T(ƒ(x)) = f'(x) + ƒ"(x). (d) T: M2x2 → M2x2 defined by T(A) = AT.

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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1 part c Solve it by: I)Creating a matrix representation Then use the characteristic polynomial to solve for lamda Then Eigenspace use (A-lamda I
1. For each of linear transformation T given below, do the following: (i) find all eigenvalues of T, (ii) find each eigenspace of T and its basis, (iii) determine the algebraic and geometric multiplicities of each eigenvalue of T, (iv) determine if T is diagonalizable. (a) T: R² R2 defined by T(a, b) = (-2a + 3b, -10a + 9b). (b) T: P₁ → P₁ defined by T(ax + b) = (-6a + 2b)x + (−6a + b). (c) T: P3 → P3 defined by T(f(x)) = f'(x) + f"(x). (d) T: M2x2 → M2x2 defined by T(A) = AT.
Transcribed Image Text:1. For each of linear transformation T given below, do the following: (i) find all eigenvalues of T, (ii) find each eigenspace of T and its basis, (iii) determine the algebraic and geometric multiplicities of each eigenvalue of T, (iv) determine if T is diagonalizable. (a) T: R² R2 defined by T(a, b) = (-2a + 3b, -10a + 9b). (b) T: P₁ → P₁ defined by T(ax + b) = (-6a + 2b)x + (−6a + b). (c) T: P3 → P3 defined by T(f(x)) = f'(x) + f"(x). (d) T: M2x2 → M2x2 defined by T(A) = AT.
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