3. Let n ≥ 1 be an integer, and let J : Pn → Pn+1 be the indefinite integration transformation on Pn, such that, for all p = P₁, the image J(p) is defined by J(p)[x] = f*p(t)dt, for all x € R. For each 0 ≤ i ≤ n + 1, define p; : R → R by p;(x) = x², and consider the bases B₂ = {Po, ...,Pn} and B₁+1 = {Po, ..,Pn+1} for P₁ and Pn+1, respectively. (a) For each p € B₁, express J(p) as a linear combination of the elements of the basis Bn+1. (b) Find the matrix representation [J]Bn+1,Bµ* (c) Find the rank and nullity of J.
3. Let n ≥ 1 be an integer, and let J : Pn → Pn+1 be the indefinite integration transformation on Pn, such that, for all p = P₁, the image J(p) is defined by J(p)[x] = f*p(t)dt, for all x € R. For each 0 ≤ i ≤ n + 1, define p; : R → R by p;(x) = x², and consider the bases B₂ = {Po, ...,Pn} and B₁+1 = {Po, ..,Pn+1} for P₁ and Pn+1, respectively. (a) For each p € B₁, express J(p) as a linear combination of the elements of the basis Bn+1. (b) Find the matrix representation [J]Bn+1,Bµ* (c) Find the rank and nullity of J.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3. Let n ≥ 1 be an integer, and let J: Pn → Pn+1 be the indefinite
integration transformation on Pn, such that, for all p = P₁, the image J(p)
E
is defined by
x
J(P) [x] = S™ P
p(t)dt, for all x € R.
For each 0 ≤ i ≤ n + 1, define p; : R → R by p;(x) = x¹, and consider
the bases B₁ {Po,,Pn} and B₁+1 {Po, ...,Pn+1} for P₁ and Pn+1,
=
=
.../
respectively.
(a) For each p E B₁, express J(p) as a linear combination of the elements
of the basis Bn+1.
(b) Find the matrix representation [J]B₁+1,B₂ °
(c) Find the rank and nullity of J.
(d) Is J surjective? Is J injective? Explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2c15801-e187-45c4-b590-39f68325995f%2F6a38e1b9-d220-4b5e-b7e8-dbbb500800d3%2Fkod1pc_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let n ≥ 1 be an integer, and let J: Pn → Pn+1 be the indefinite
integration transformation on Pn, such that, for all p = P₁, the image J(p)
E
is defined by
x
J(P) [x] = S™ P
p(t)dt, for all x € R.
For each 0 ≤ i ≤ n + 1, define p; : R → R by p;(x) = x¹, and consider
the bases B₁ {Po,,Pn} and B₁+1 {Po, ...,Pn+1} for P₁ and Pn+1,
=
=
.../
respectively.
(a) For each p E B₁, express J(p) as a linear combination of the elements
of the basis Bn+1.
(b) Find the matrix representation [J]B₁+1,B₂ °
(c) Find the rank and nullity of J.
(d) Is J surjective? Is J injective? Explain your answer.
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