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 +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(ƒ(x)) = ïƒ'(x) + f(2)x+ ƒ(3) (f) V=P3(R) and T(ƒ(x)) = f(x) + f(2)x (g) V=P3(R) and T(ƒ(x)) = xƒ'(x) + ƒ"(x) − ƒ(2)
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 +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(ƒ(x)) = ïƒ'(x) + f(2)x+ ƒ(3) (f) V=P3(R) and T(ƒ(x)) = f(x) + f(2)x (g) V=P3(R) and T(ƒ(x)) = xƒ'(x) + ƒ"(x) − ƒ(2)
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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Just answer b,f,g
![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 +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(ƒ(x)) = ïƒ'(x) + ƒ(2)x+ ƒ(3)
(f) V=P3(R) and T(f(x)) = f(x) + f(2)x
(g) V=P3(R) and T(ƒ(x)) = xƒ'(x) + ƒ"(x) − ƒ(2)
(h) V=M₂x2 (R) and T
=
с
=
(i) V = M₂x2(R) and T
a
(j) V=M₂x2(R) and T(A) = A' + 2 · tr(A) - 12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F327bc9af-4fd4-4b75-a0b4-6f6a68972d6f%2Fjpoige_processed.jpeg&w=3840&q=75)
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]; 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(ƒ(x)) = ïƒ'(x) + ƒ(2)x+ ƒ(3)
(f) V=P3(R) and T(f(x)) = f(x) + f(2)x
(g) V=P3(R) and T(ƒ(x)) = xƒ'(x) + ƒ"(x) − ƒ(2)
(h) V=M₂x2 (R) and T
=
с
=
(i) V = M₂x2(R) and T
a
(j) V=M₂x2(R) and T(A) = A' + 2 · tr(A) - 12
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