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Problem 8. Consider the inner product space from Problem 4. Find an orthonormal basis for the subspace of C[-1,1] spanned by functions h₁(x) = 1, h₂(x) = x and h3(x) = x².

Problem 8. Consider the inner product space from Problem 4. Find an orthonormal basis for the subspace of C[-1,1] spanned by functions h₁(x) = 1, h₂(x) = x and h3(x) = x².

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 3AEXP
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Please solve problem 8 with reference to problem 4

Problem 4. Consider the inner product space C[-1, 1] with the inner product defined by (f,g) = f(x)g(x) dx and the induced norm. Find the best least squares approximation to the function f(x) = x¹/³ on [-1, 1] by a linear function ((x) = ₁ + ₂x. [Hint: first show that the functions h₁(x) = 1 and h₂(x) = x are orthogonal.]
Transcribed Image Text:Problem 4. Consider the inner product space C[-1, 1] with the inner product defined by (f,g) = f(x)g(x) dx and the induced norm. Find the best least squares approximation to the function f(x) = x¹/³ on [-1, 1] by a linear function ((x) = ₁ + ₂x. [Hint: first show that the functions h₁(x) = 1 and h₂(x) = x are orthogonal.]
Problem 8. Consider the inner product space from Problem 4. Find an orthonormal basis for the subspace of C[-1,1] spanned by functions h₁(x) = 1, h₂(x) = x and h3(x) = x².
Transcribed Image Text:Problem 8. Consider the inner product space from Problem 4. Find an orthonormal basis for the subspace of C[-1,1] spanned by functions h₁(x) = 1, h₂(x) = x and h3(x) = x².
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