1. Consider the inner product space (C[-1, 1], (,)), where we assume we are working with the standard L² inner product. Let 1, 2, 3, ƒ E C[−1, 1] be defined by 01(x) = 1, 02(x) = x, 03(x) = x², and f(x) = e. Construct the best approximation of f from the following subspaces of C[-1, 1]. (a) Construct the best approximation, denoted fi, to f from W₁ = span {#1}. (b) Construct the best approximation, f2, to f from W₂ = span {1, 2}. (c) Construct the best approximation, f3, to f from W3 = span {01, 02, 03}. (d) Produce a plot that superimposes your best approximations from parts (a), (b), and (c) on top of a plot of f(x). Give an interpretation of your results. Does the projection get better when you increase the number of basis functions in the finite dimensional subspace you project onto? Why or why not?
1. Consider the inner product space (C[-1, 1], (,)), where we assume we are working with the standard L² inner product. Let 1, 2, 3, ƒ E C[−1, 1] be defined by 01(x) = 1, 02(x) = x, 03(x) = x², and f(x) = e. Construct the best approximation of f from the following subspaces of C[-1, 1]. (a) Construct the best approximation, denoted fi, to f from W₁ = span {#1}. (b) Construct the best approximation, f2, to f from W₂ = span {1, 2}. (c) Construct the best approximation, f3, to f from W3 = span {01, 02, 03}. (d) Produce a plot that superimposes your best approximations from parts (a), (b), and (c) on top of a plot of f(x). Give an interpretation of your results. Does the projection get better when you increase the number of basis functions in the finite dimensional subspace you project onto? Why or why not?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.5: Subspaces, Basis, Dimension, And Rank
Problem 10EQ
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![1. Consider the inner product space (C[-1, 1], (,)), where we assume we are working with the standard
L² inner product. Let 1, 2, 3, ƒ E C[−1, 1] be defined by
01(x) = 1, 02(x) = x, 03(x) = x²,
and
f(x) = e.
Construct the best approximation of f from the following subspaces of C[-1, 1].
(a) Construct the best approximation, denoted fi, to f from W₁ = span {#1}.
(b) Construct the best approximation, f2, to f from W₂ = span {1, 2}.
(c) Construct the best approximation, f3, to f from W3 = span {01, 02, 03}.
(d) Produce a plot that superimposes your best approximations from parts (a), (b), and (c) on top
of a plot of f(x). Give an interpretation of your results. Does the projection get better when you
increase the number of basis functions in the finite dimensional subspace you project onto? Why
or why not?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9328926a-b343-4d47-8a00-cbd9e6e36ac2%2F029f8fb1-1f1f-4c58-9c4d-282bbb99d66f%2F0ui7utc_processed.png&w=3840&q=75)
Transcribed Image Text:1. Consider the inner product space (C[-1, 1], (,)), where we assume we are working with the standard
L² inner product. Let 1, 2, 3, ƒ E C[−1, 1] be defined by
01(x) = 1, 02(x) = x, 03(x) = x²,
and
f(x) = e.
Construct the best approximation of f from the following subspaces of C[-1, 1].
(a) Construct the best approximation, denoted fi, to f from W₁ = span {#1}.
(b) Construct the best approximation, f2, to f from W₂ = span {1, 2}.
(c) Construct the best approximation, f3, to f from W3 = span {01, 02, 03}.
(d) Produce a plot that superimposes your best approximations from parts (a), (b), and (c) on top
of a plot of f(x). Give an interpretation of your results. Does the projection get better when you
increase the number of basis functions in the finite dimensional subspace you project onto? Why
or why not?
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