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2. Let P, denote the space of polynomials of degreek in z, and consider D: P P. Dp(x) -p'(x). Show that D1-0 on P, and that (1,2,...) is a basis of P, with respect to which D is strictly upper triangular.

2. Let P, denote the space of polynomials of degreek in z, and consider D: P P. Dp(x) -p'(x). Show that D1-0 on P, and that (1,2,...) is a basis of P, with respect to which D is strictly upper triangular.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.4: Zeros Of A Polynomial
Problem 22E
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92 of 383 1. Consider /0 0 4- (121). A₂01 mtaylor.web.unc.edu Exercises (9) 0 32 Compute the characteristic polynomial of each A, and verify that these matrices satisfy the Caley-Hamilton theorem, (2.3.13). 2. Let P, denote the space of polynomials of degree ≤k in z, and consider D: PPk Dp(z) = p'(x). Show that D²+1 = 0 on P, and that {1,2,...,} is a basis of P, with respect to which D is strictly upper triangular. 3. Use the identity to obtain a solution u € P to (2.3.25) to obtain a formula for Exercises (I-D)-¹-D, on P 4. Use the equivalence of (2.3.25) with A₂- -u-z Show that [zez dz. 5. The proof of Proposition 2.3.1 given above includes the chain of implica- tions (2.3.4)→ (2.3.2) ↔ (2.3.3) → (2.3.4). + Use Proposition 2.2.4 to give another proof that (2.3.3) (2.3.2). → 6. Establish the following variant of Proposition 2.2.4. Let K(A) be the characteristic polynomial of T, as in (2.3.12), and set P(A) - II(A-A₂)²(A-Aede Κτ(λ) 344 77 GE(T) = R(P(T)). 7. Show that, if X, is a root of det (AI-A) = 0 of multiplicity dj, then dim GE (A, Aj) = d., and GE(A, X) = N((A-X,1)). For a refinement of the latter identity, see Exercise 4 in the sext section.
Transcribed Image Text:92 of 383 1. Consider /0 0 4- (121). A₂01 mtaylor.web.unc.edu Exercises (9) 0 32 Compute the characteristic polynomial of each A, and verify that these matrices satisfy the Caley-Hamilton theorem, (2.3.13). 2. Let P, denote the space of polynomials of degree ≤k in z, and consider D: PPk Dp(z) = p'(x). Show that D²+1 = 0 on P, and that {1,2,...,} is a basis of P, with respect to which D is strictly upper triangular. 3. Use the identity to obtain a solution u € P to (2.3.25) to obtain a formula for Exercises (I-D)-¹-D, on P 4. Use the equivalence of (2.3.25) with A₂- -u-z Show that [zez dz. 5. The proof of Proposition 2.3.1 given above includes the chain of implica- tions (2.3.4)→ (2.3.2) ↔ (2.3.3) → (2.3.4). + Use Proposition 2.2.4 to give another proof that (2.3.3) (2.3.2). → 6. Establish the following variant of Proposition 2.2.4. Let K(A) be the characteristic polynomial of T, as in (2.3.12), and set P(A) - II(A-A₂)²(A-Aede Κτ(λ) 344 77 GE(T) = R(P(T)). 7. Show that, if X, is a root of det (AI-A) = 0 of multiplicity dj, then dim GE (A, Aj) = d., and GE(A, X) = N((A-X,1)). For a refinement of the latter identity, see Exercise 4 in the sext section.
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