= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).
= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).
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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4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ =
{ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by
Define T: P2 P₂ by
(p\q) = p(x)q(x) dx.
0
T(ao + ax + a2x²) = a₁x.
(a) Show that T is not Hermitian (i.e. self-adjoint).
(b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian.
Explain why this does not contradict with result from Part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F655e5d67-ab19-404c-b883-3aa3c693f6a1%2Ffea782ff-73ae-44da-8ee7-acb04bff779c%2F9s4aynp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:=
4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ =
{ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by
Define T: P2 P₂ by
(p\q) = p(x)q(x) dx.
0
T(ao + ax + a2x²) = a₁x.
(a) Show that T is not Hermitian (i.e. self-adjoint).
(b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian.
Explain why this does not contradict with result from Part (a).
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