(a) Show that the standard basis {1, x, x^2} is not orthogonal with respect to this inner product.(b) (15) Use the standard basis {1, x, x^2} to find an orthonormal basis for this inner product space. "5. Consider the space \( P_2 \) (the space of polynomials of degree at most 2) and define the inner product\[\langle p, q \rangle = \frac{1}{3} \left( p(0)q(0) + p(1)q(1) + p(2)q(2) \right).\]"

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Answered: 5. Consider the space P2 (the space of…

5. Consider the space P2 (the space of polynomials of degree at most 2) and define the inner product 1 (p, q) = = (p(0)q(0) + p(1)q(1) + p(2)q(2)).

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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Question

(a) Show that the standard basis {1, x, x^2} is not orthogonal with respect to this inner product.
(b) (15) Use the standard basis {1, x, x^2} to find an orthonormal basis for this inner product space.

"5. Consider the space \( P_2 \) (the space of polynomials of degree at most 2) and define the inner product

\[
\langle p, q \rangle = \frac{1}{3} \left( p(0)q(0) + p(1)q(1) + p(2)q(2) \right).
\]"

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