4. Consider the vector space C[-1, 1] of continuous functions on [-1, 1] with inner product (f,g) = [*, f(x)g(x) dx. (a) Show that the vectors 1 and x are orthogonal. (b) Compute ||1|| and || x ||. (c) Find the best least squares approximation to x¹/3 on [1,1] by a linear function 1(x) = c₁1+c₂x.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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4. Consider the vector space C[-1, 1] of continuous functions on [1,1] with inner product
(f.g) = [ f(x)g(x) dx.
(a) Show that the vectors 1 and x are orthogonal.
(b) Compute || 1 || and || x ||.
=
(c) Find the best least squares approximation to x¹/3 on [-1,1] by a linear function (x)
c₁1+c₂x.
Transcribed Image Text:4. Consider the vector space C[-1, 1] of continuous functions on [1,1] with inner product (f.g) = [ f(x)g(x) dx. (a) Show that the vectors 1 and x are orthogonal. (b) Compute || 1 || and || x ||. = (c) Find the best least squares approximation to x¹/3 on [-1,1] by a linear function (x) c₁1+c₂x.
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