21. Show that the set of polynomials {C₁, C₂ (2x - 1), C3 (6x² − 6x + 1)} in P(R) is orthogonal with respect to the inner product f f(x) g(x) dx for any choice of the constants C₁, C2, C3. For what values of C₁, C2, C3 is the set orthonormal?
21. Show that the set of polynomials {C₁, C₂ (2x - 1), C3 (6x² − 6x + 1)} in P(R) is orthogonal with respect to the inner product f f(x) g(x) dx for any choice of the constants C₁, C2, C3. For what values of C₁, C2, C3 is the set orthonormal?
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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![21. Show that the set of polynomials
\[
\{c_1, c_2(2x - 1), c_3(6x^2 - 6x + 1)\}
\]
in \( P(\mathbb{R}) \) is orthogonal with respect to the inner product \(\int_0^1 f(x) \, g(x) \, dx\) for any choice of the constants \( c_1, c_2, c_3 \). For what values of \( c_1, c_2, c_3 \) is the set orthonormal?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2Fb93fb514-767b-4577-87c5-116c21fc8358%2Fxcy2vzk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:21. Show that the set of polynomials
\[
\{c_1, c_2(2x - 1), c_3(6x^2 - 6x + 1)\}
\]
in \( P(\mathbb{R}) \) is orthogonal with respect to the inner product \(\int_0^1 f(x) \, g(x) \, dx\) for any choice of the constants \( c_1, c_2, c_3 \). For what values of \( c_1, c_2, c_3 \) is the set orthonormal?
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