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3) Consider the weighted inner product {p,q) = fo° p(x)q(x)e¯* dx on C[0, ∞) the space of functions continuous on the interval [0, ∞). Use Mathemat- ica to answer the following problems. a) Construct an orthonormal set from the set {1, x, x², x³, x², x³}. b) Use the orthonormal set to find an approximation to sin(x). Give the coefficients numerically. c) Plot sin(x) and the approximation from b.) on the same set of axes on the interval 0 ≤ x ≤ 10. d) Compute the metric distance between sin(x) and the approximation from b.). e) Why are the graphs so different despite the distance between the functions being so small?

3) Consider the weighted inner product {p,q) = fo° p(x)q(x)e¯* dx on C[0, ∞) the space of functions continuous on the interval [0, ∞). Use Mathemat- ica to answer the following problems. a) Construct an orthonormal set from the set {1, x, x², x³, x², x³}. b) Use the orthonormal set to find an approximation to sin(x). Give the coefficients numerically. c) Plot sin(x) and the approximation from b.) on the same set of axes on the interval 0 ≤ x ≤ 10. d) Compute the metric distance between sin(x) and the approximation from b.). e) Why are the graphs so different despite the distance between the functions being so small?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 15EQ
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3) Consider the weighted inner product {p,q) = fo° p(x)q(x)e¯* dx on C[0, ∞) the space of functions continuous on the interval [0, ∞). Use Mathemat- ica to answer the following problems. a) Construct an orthonormal set from the set {1, x, x², x³, x², x³}. b) Use the orthonormal set to find an approximation to sin(x). Give the coefficients numerically. c) Plot sin(x) and the approximation from b.) on the same set of axes on the interval 0 ≤ x ≤ 10. d) Compute the metric distance between sin(x) and the approximation from b.). e) Why are the graphs so different despite the distance between the functions being so small?
Transcribed Image Text:3) Consider the weighted inner product {p,q) = fo° p(x)q(x)e¯* dx on C[0, ∞) the space of functions continuous on the interval [0, ∞). Use Mathemat- ica to answer the following problems. a) Construct an orthonormal set from the set {1, x, x², x³, x², x³}. b) Use the orthonormal set to find an approximation to sin(x). Give the coefficients numerically. c) Plot sin(x) and the approximation from b.) on the same set of axes on the interval 0 ≤ x ≤ 10. d) Compute the metric distance between sin(x) and the approximation from b.). e) Why are the graphs so different despite the distance between the functions being so small?
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