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Get Full Access to Linear Algebra - 4 Edition - Chapter 7.3 - Problem 11
Get Full Access to Linear Algebra - 4 Edition - Chapter 7.3 - Problem 11

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# Let g(t) be the auxiliary polynomial associated with a homogeneous linear differential ISBN: 9780130084514 53

## Solution for problem 11 Chapter 7.3

Linear Algebra | 4th Edition

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Problem 11

Let g(t) be the auxiliary polynomial associated with a homogeneous linear differential equation with constant coefficients (as defined in Section 2.7), and let V denote the solution space of this differential equation. Prove the following results. (a) V is a D-invariant subspace, where D is the differentiation operator on C. (b) The minimal polynomial of Dv (the restriction of D to V) is g(t). (c) If the degree of g(t) is n, then the characteristic polynomial of Dv is(-l)n g(t). Hint: Use Theorem 2.32 (p. 135) for (b) and (c).

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##### ISBN: 9780130084514

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