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5. For each linear operator T on V, find the eigenvalues of T and an ordered basis for V such that [T] is a diagonal matrix. (a) V=R2 and T(a, b) = (-2a + 3b, -10a +96) (b) V = R³ and T(a, b, c) = (7a-4b + 10c, 4a − 3b + 8c, −2a + b - 2c) (c) V = R³ and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a-6b+11c) (d) V = P₁(R) and T(ax + b) = (-6a + 2b)x+ (-6a+b) (e) V = P₂(R) and T(f(x)) = xf'(x) + f(2)x+ f(3) (f) V=P3(R) and T(f(x)) = f(x) + f(2)x (g) V=P3(R) and T(f(x)) = xf'(x) + f"(x) = f(2) (h) V=M2x2 (R) and T - (ad) = (dd) a (dd) = (d) C a (j) V = M2x2 (R) and T(A) = A + 2. tr(A) - 1₂ (i) V = M2x2 (R) and T

5. For each linear operator T on V, find the eigenvalues of T and an ordered basis for V such that [T] is a diagonal matrix. (a) V=R2 and T(a, b) = (-2a + 3b, -10a +96) (b) V = R³ and T(a, b, c) = (7a-4b + 10c, 4a − 3b + 8c, −2a + b - 2c) (c) V = R³ and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a-6b+11c) (d) V = P₁(R) and T(ax + b) = (-6a + 2b)x+ (-6a+b) (e) V = P₂(R) and T(f(x)) = xf'(x) + f(2)x+ f(3) (f) V=P3(R) and T(f(x)) = f(x) + f(2)x (g) V=P3(R) and T(f(x)) = xf'(x) + f"(x) = f(2) (h) V=M2x2 (R) and T - (ad) = (dd) a (dd) = (d) C a (j) V = M2x2 (R) and T(A) = A + 2. tr(A) - 1₂ (i) V = M2x2 (R) and T

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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5. For each linear operator T on V, find the eigenvalues of T and an ordered basis for V such that [T]s is a diagonal matrix. (a) V=R2 and T(a, b) = (-2a + 3b, -10a +9b) - (b) V = R³ and T(a, b, c) = (7a-4b + 10c, 4a − 3b+8c, -2a+b-2c) (c) V = R³ and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a-6b+11c) (d) V = P₁(R) and T(ax + b) = (-6a + 2b)x+ (-6a+b) (e) V = P₂(R) and T(f(x)) = f'(x) + f(2)x+ f(3) (f) V=P3(R) and T(f(x)) = f(x) + f(2)x (g) V = P3(R) and T(f(x)) = xƒ'(x) + ƒ"(x) = f(2) C - (ad) = (dà) (cd) = (3) a C a (j) V = M2x2 (R) and T(A) = A + 2. tr(A) - 1₂ (h) V=M2x2 (R) and T (i) V = M₂x2 (R) and T
Transcribed Image Text:5. For each linear operator T on V, find the eigenvalues of T and an ordered basis for V such that [T]s is a diagonal matrix. (a) V=R2 and T(a, b) = (-2a + 3b, -10a +9b) - (b) V = R³ and T(a, b, c) = (7a-4b + 10c, 4a − 3b+8c, -2a+b-2c) (c) V = R³ and T(a, b, c) = (-4a+3b-6c, 6a-7b+12c, 6a-6b+11c) (d) V = P₁(R) and T(ax + b) = (-6a + 2b)x+ (-6a+b) (e) V = P₂(R) and T(f(x)) = f'(x) + f(2)x+ f(3) (f) V=P3(R) and T(f(x)) = f(x) + f(2)x (g) V = P3(R) and T(f(x)) = xƒ'(x) + ƒ"(x) = f(2) C - (ad) = (dà) (cd) = (3) a C a (j) V = M2x2 (R) and T(A) = A + 2. tr(A) - 1₂ (h) V=M2x2 (R) and T (i) V = M₂x2 (R) and T
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