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5. For each linear operator T on V, find the eigenvalues of T and an ordered basis ẞ for V such that [T] is a diagonal matrix. (a) VR2 and T(a, b) = (-2a+3b,-10a + 9b) (b) VR3 and T(a, b, c) = (7a-4b+ 10c, 4a - 36+ 8c, -2a+b2c) (c) VR3 and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a6b+11c) (d) V P₁(R) and T(ax + b) = (-6a+ 2b)x+(-6a+b) = (e) VP2(R) and T(f(x)) = xf'(x) + f(2)x+ƒ(3) (f) VP3(R) and T(f(x)) = f(x) + f(2)x (g) VP3(R) and T(f(x)) = xf'(x) + f'(x) − f(2) (h) VM2x2(R) and T (i) V M2x2(R) and T = ( a b ) = ( a b ) 9) (d b) (a b) = ( c d ) (j) V=M2x2(R) and T(A) = A +2.tr(A) 12

5. For each linear operator T on V, find the eigenvalues of T and an ordered basis ẞ for V such that [T] is a diagonal matrix. (a) VR2 and T(a, b) = (-2a+3b,-10a + 9b) (b) VR3 and T(a, b, c) = (7a-4b+ 10c, 4a - 36+ 8c, -2a+b2c) (c) VR3 and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a6b+11c) (d) V P₁(R) and T(ax + b) = (-6a+ 2b)x+(-6a+b) = (e) VP2(R) and T(f(x)) = xf'(x) + f(2)x+ƒ(3) (f) VP3(R) and T(f(x)) = f(x) + f(2)x (g) VP3(R) and T(f(x)) = xf'(x) + f'(x) − f(2) (h) VM2x2(R) and T (i) V M2x2(R) and T = ( a b ) = ( a b ) 9) (d b) (a b) = ( c d ) (j) V=M2x2(R) and T(A) = A +2.tr(A) 12

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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Assignment 9 Section 5.1:Number 5(d) only! perfectly and complete explanations.

5. For each linear operator T on V, find the eigenvalues of T and an ordered basis ẞ for V such that [T] is a diagonal matrix. (a) VR2 and T(a, b) = (-2a+3b,-10a + 9b) (b) VR3 and T(a, b, c) = (7a-4b+ 10c, 4a - 36+ 8c, -2a+b2c) (c) VR3 and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a6b+11c) (d) V P₁(R) and T(ax + b) = (-6a+ 2b)x+(-6a+b) = (e) VP2(R) and T(f(x)) = xf'(x) + f(2)x+ƒ(3) (f) VP3(R) and T(f(x)) = f(x) + f(2)x (g) VP3(R) and T(f(x)) = xf'(x) + f'(x) − f(2) (h) VM2x2(R) and T (i) V M2x2(R) and T = ( a b ) = ( a b ) 9) (d b) (a b) = ( c d ) (j) V=M2x2(R) and T(A) = A +2.tr(A) 12
Transcribed Image Text:5. For each linear operator T on V, find the eigenvalues of T and an ordered basis ẞ for V such that [T] is a diagonal matrix. (a) VR2 and T(a, b) = (-2a+3b,-10a + 9b) (b) VR3 and T(a, b, c) = (7a-4b+ 10c, 4a - 36+ 8c, -2a+b2c) (c) VR3 and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a6b+11c) (d) V P₁(R) and T(ax + b) = (-6a+ 2b)x+(-6a+b) = (e) VP2(R) and T(f(x)) = xf'(x) + f(2)x+ƒ(3) (f) VP3(R) and T(f(x)) = f(x) + f(2)x (g) VP3(R) and T(f(x)) = xf'(x) + f'(x) − f(2) (h) VM2x2(R) and T (i) V M2x2(R) and T = ( a b ) = ( a b ) 9) (d b) (a b) = ( c d ) (j) V=M2x2(R) and T(A) = A +2.tr(A) 12
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