#3 92 of 3831. Consider/0 04- (121). A₂01mtaylor.web.unc.eduExercises(9)032Compute the characteristic polynomial of each A, and verify that thesematrices satisfy the Caley-Hamilton theorem, (2.3.13).2. Let P, denote the space of polynomials of degree ≤k in z, and considerD: PPk Dp(z) = p'(x).Show that D²+1 = 0 on P, and that {1,2,...,} is a basis of P, withrespect to which D is strictly upper triangular.3. Use the identityto obtain a solution u € P to(2.3.25)to obtain a formula forExercises(I-D)-¹-D, on P4. Use the equivalence of (2.3.25) withA₂--u-zShow 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 thecharacteristic polynomial of T, as in (2.3.12), and setP(A) - II(A-A₂)²(A-AedeΚτ(λ)34477GE(T) = R(P(T)).7. Show that, if X, is a root of det (AI-A) = 0 of multiplicity dj, thendim 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.

Answered: 92 of 383 1. Consider…

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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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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What is Basis of a Vector Space:

The set B of vectors in the vector space V is referred to as a basis in mathematics if every member of V can be written in a unique way as a finite linear combination of components of B. The coefficients of this linear combination are the elements, or coordinates, of the vector with respect to B. Basis components are referred to as basis vectors. A set B is a basis if all of its members are linearly independent and every element of V is a linear combination of all of its members. A basis is a spanning set that is linearly independent, to put it another way.

To Determine:

We determine a solution $u \in P_{k}$ satisfying the differential equation $u' - u = x^{k}$ .

Here, $P_{k}$ is the space of all polynomials having degree less than or equal to $k over x$ . The basis of $k over x$ is

$B = \{1, x, x^{2}, . . . ., x^{k}\}$ .