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T be a linear operator on a finite-dimensional vector space V, and let Wi be a

ISBN: 9780130084514 53

Solution for problem 16 Chapter 7.3

Linear Algebra | 4th Edition

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Problem 16

T be a linear operator on a finite-dimensional vector space V, and let Wi be a T-invariant subspace of V. Let x G V such that .; ^ Wi. Prove the following results. (a) There exists a unique monic polynomial g\ (t) of least positive degree such that 0] (T)(.T) G WI. (b) If fi.(t) is a polynomial for which h(T)(x) G Wi, then gi(t) divides h(t). (c) gi(t) divides the minimal and the characteristic polynomials of T. (d) Let W2 be a T-invariant subspace of V such that W2 C W^ and let g2(t) be the unique monic polynomial of least degree such that g2(T)(x) G W2. Then gi(t) divides g2(t).

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Week 13 Monday, November 7, 20112:00 PM Tuesday, November 8, 2016 12:36 PM Tuesday, November 8, 2016 12:46 PM Wednesday, November 9, 2016 12:11 PM Wednesday, November 9, 2016 12:17 PM Wednesday, November 9, 2016 12:36 PM Wednesday, November 9, 2016 12:38 PM

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T be a linear operator on a finite-dimensional vector space V, and let Wi be a