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Orthonormalize [1, t, t²} in P₂(R) with (p, q) = p(0)q(0) + p(1)q(1) +p(2)q(2) }

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
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.
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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