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(b) Let B = (1, x, x², x³) be the standard basis for cubic polynomials (P3) and define an inner product for f, g € P3 as (f,g) = [*¸ ƒ(x)g(x)dx i. For a non-negative integer k, evaluate the following integral to show that 2 [₁₁2²dx = { 1 + k+1 k even k odd ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis C = (ho, h1, h2, h3) for P3.

(b) Let B = (1, x, x², x³) be the standard basis for cubic polynomials (P3) and define an inner product for f, g € P3 as (f,g) = [*¸ ƒ(x)g(x)dx i. For a non-negative integer k, evaluate the following integral to show that 2 [₁₁2²dx = { 1 + k+1 k even k odd ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis C = (ho, h1, h2, h3) for P3.

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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part b)

 

(b) Let B = (1, x, xr², x³) be the standard basis for cubic polynomials (P3) and define an inner product for f, g = P3 as (f.g) = [ f(x)g(x)dx i. For a non-negative integer k, evaluate the following integral to show that L at der = {² dx 1 even k odd ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis C = (ho, h1, h2, h3) for P3.
Transcribed Image Text:(b) Let B = (1, x, xr², x³) be the standard basis for cubic polynomials (P3) and define an inner product for f, g = P3 as (f.g) = [ f(x)g(x)dx i. For a non-negative integer k, evaluate the following integral to show that L at der = {² dx 1 even k odd ii. Apply the Gram-Schmidt algorithm on B (in the given order) to find an orthogonal basis C = (ho, h1, h2, h3) for P3.
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