Consider the matrix A = substituting z = A: (8) 0 Define an "evaluation" homomorphism: R[x] - → M₂ (R) by (anx + an-1xn-1. +...+ a₁x+ao) Here the powers A denote the usual powers of the matrix A under matrix multiplication, and I = denotes the identity matrix. (² 0 You do not have to prove that is a homomorphism, but I encourage you to think about why this is true. (a) Compute A². What is Ak for k ≥ 3? (b) Show that the image of the homomorphism is: = an A" + an-1A−¹ +...+ a₁A+aoI ao 0 b-{(2) man} Im(x) = ao, a₁ € R R} a1 ao (c) The characteristic polynomial of A is g(x) = x² of A. Show that g € Ker(p). (d) Issurjective? Injective? 0 1
Consider the matrix A = substituting z = A: (8) 0 Define an "evaluation" homomorphism: R[x] - → M₂ (R) by (anx + an-1xn-1. +...+ a₁x+ao) Here the powers A denote the usual powers of the matrix A under matrix multiplication, and I = denotes the identity matrix. (² 0 You do not have to prove that is a homomorphism, but I encourage you to think about why this is true. (a) Compute A². What is Ak for k ≥ 3? (b) Show that the image of the homomorphism is: = an A" + an-1A−¹ +...+ a₁A+aoI ao 0 b-{(2) man} Im(x) = ao, a₁ € R R} a1 ao (c) The characteristic polynomial of A is g(x) = x² of A. Show that g € Ker(p). (d) Issurjective? Injective? 0 1
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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Could you solve (c) and (d)?
Thank you.
![4. Consider the matrix A =
substituting ax = A:
0 0
(8)
1
(anx + an-11²
Define an "evaluation" homomorphism: R[x] - → M₂ (R) by
g.n-1 + + a₁x + ao) = an An+an-1 An-1 +... . + a₁A+aoI
Here the powers A denote the usual powers of the matrix A under matrix multiplication, and I =
(69)
denotes the identity matrix.
You do not have to prove that is a homomorphism, but I encourage you to think about why this is true.
(a) Compute A2. What is Ak for k≥ 3?
(b) Show that the image of the homomorphism & is:
= {(2 %)
a1 ao
Im(y) =
: ao, a₁ ER
(c) The characteristic polynomial of A is g(x) = x² of A. Show that g = Ker(y).
(d) Issurjective? Injective?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2F40de3cbc-683e-4f3c-88ff-7f858becb3f6%2Fyvsr6mw_processed.png&w=3840&q=75)
Transcribed Image Text:4. Consider the matrix A =
substituting ax = A:
0 0
(8)
1
(anx + an-11²
Define an "evaluation" homomorphism: R[x] - → M₂ (R) by
g.n-1 + + a₁x + ao) = an An+an-1 An-1 +... . + a₁A+aoI
Here the powers A denote the usual powers of the matrix A under matrix multiplication, and I =
(69)
denotes the identity matrix.
You do not have to prove that is a homomorphism, but I encourage you to think about why this is true.
(a) Compute A2. What is Ak for k≥ 3?
(b) Show that the image of the homomorphism & is:
= {(2 %)
a1 ao
Im(y) =
: ao, a₁ ER
(c) The characteristic polynomial of A is g(x) = x² of A. Show that g = Ker(y).
(d) Issurjective? Injective?
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