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Problem 3: Let T : V → V be a linear map, let v € V, and let W be the T-cyclic subspace of V generated by v. a.) For any u € V, show that u € W if and only if there exists a polynomial f(t) such that u = f(T)v. b.) Suppose dim W = k. Show that f(t) can be taken to have degree at most k. Problem 4: Let T : V → V be a linear map. Suppose V has a T-cyclic vector, that is, suppose there exists v € V such that the T-cyclic subspace generated by v is equal to all of V. If S : V → V is any linear map, then prove that ST = TS if and only if S = f(T) for some polynomial f(t). (Hint: Take u = S(v), and apply the result in problem 3(a).)

Problem 3: Let T : V → V be a linear map, let v € V, and let W be the T-cyclic subspace of V generated by v. a.) For any u € V, show that u € W if and only if there exists a polynomial f(t) such that u = f(T)v. b.) Suppose dim W = k. Show that f(t) can be taken to have degree at most k. Problem 4: Let T : V → V be a linear map. Suppose V has a T-cyclic vector, that is, suppose there exists v € V such that the T-cyclic subspace generated by v is equal to all of V. If S : V → V is any linear map, then prove that ST = TS if and only if S = f(T) for some polynomial f(t). (Hint: Take u = S(v), and apply the result in problem 3(a).)

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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only question 4

(Hint: Take u = S(v), and apply the result in problem 3(a).)

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Problem 3: Let T: V → V be a linear map, let v € V, and let W be the T-cyclic subspace of V generated by v. a.) For any u € V, show that u € W if and only if there exists a polynomial f(t) such that u = f(T)v. b.) Suppose dim W = k. Show that f(t) can be taken to have degree at most k. Problem 4: Let T : V → V be a linear map. Suppose V has a T-cyclic vector, that is, suppose there exists v € V such that the T-cyclic subspace generated by v is equal to all of V. If S : V → V is any linear map, then prove that ST = TS if and only if S = f(T) for some polynomial f(t). (Hint: Take u = S(v), and apply the result in problem 3(a).)
Transcribed Image Text:Problem 3: Let T: V → V be a linear map, let v € V, and let W be the T-cyclic subspace of V generated by v. a.) For any u € V, show that u € W if and only if there exists a polynomial f(t) such that u = f(T)v. b.) Suppose dim W = k. Show that f(t) can be taken to have degree at most k. Problem 4: Let T : V → V be a linear map. Suppose V has a T-cyclic vector, that is, suppose there exists v € V such that the T-cyclic subspace generated by v is equal to all of V. If S : V → V is any linear map, then prove that ST = TS if and only if S = f(T) for some polynomial f(t). (Hint: Take u = S(v), and apply the result in problem 3(a).)
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