Orthonormalize [1, t, t²} in P₂(R) with (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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![**Orthogonalization of Polynomial Basis**
Objective: Orthonormalize the set \(\{1, t, t^2\}\) in \(P_2(\mathbb{R})\).
The inner product \(\langle p, q \rangle\) is defined as follows:
\[
\langle p, q \rangle = p(0)q(0) + p(1)q(1) + p(2)q(2)
\]
This requires the determination of an orthonormal basis set with respective placeholders:
\[
\left\{ \boxed{\phantom{a}} , \boxed{\phantom{a}} , \boxed{\phantom{a}} \right\}
\]
Details:
- **Orthonormalization** involves creating a set of vectors that are both orthogonal (perpendicular) to each other and normalized (having unit length).
- **Polynomials in \(P_2(\mathbb{R})\)** contain terms up to and including the square of \(t\).
- **Inner Product Calculation** involves evaluating the polynomials at specific points (0, 1, and 2) and summing the products of these evaluations for a pair of polynomials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86f92849-52f1-43f0-ad77-c1523309cdc1%2F5a12211f-5a96-4e91-b681-b4d4c08da2cd%2F4s9p6zq_processed.png&w=3840&q=75)
Transcribed Image Text:**Orthogonalization of Polynomial Basis**
Objective: Orthonormalize the set \(\{1, t, t^2\}\) in \(P_2(\mathbb{R})\).
The inner product \(\langle p, q \rangle\) is defined as follows:
\[
\langle p, q \rangle = p(0)q(0) + p(1)q(1) + p(2)q(2)
\]
This requires the determination of an orthonormal basis set with respective placeholders:
\[
\left\{ \boxed{\phantom{a}} , \boxed{\phantom{a}} , \boxed{\phantom{a}} \right\}
\]
Details:
- **Orthonormalization** involves creating a set of vectors that are both orthogonal (perpendicular) to each other and normalized (having unit length).
- **Polynomials in \(P_2(\mathbb{R})\)** contain terms up to and including the square of \(t\).
- **Inner Product Calculation** involves evaluating the polynomials at specific points (0, 1, and 2) and summing the products of these evaluations for a pair of polynomials.
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