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2. Consider the inner product space (P2(-1, 1), (,)) with inner product (f,9) f(x)g(x) dx, Vf, 9 € P2(-1, 1). (a) Consider the following basis B = {Po: P1, P2} for P2(-1, 1) with Po(x) = 1, P₁(x) = x, P2(x) = (3x² - 1). Verify that B is indeed an orthogonal set. Also, compute the norm of each element of B and conclude whether or not the set is orthonormal. (b) Using inner products, compute [q]B for q(x) = 1+x+x².

2. Consider the inner product space (P2(-1, 1), (,)) with inner product (f,9) f(x)g(x) dx, Vf, 9 € P2(-1, 1). (a) Consider the following basis B = {Po: P1, P2} for P2(-1, 1) with Po(x) = 1, P₁(x) = x, P2(x) = (3x² - 1). Verify that B is indeed an orthogonal set. Also, compute the norm of each element of B and conclude whether or not the set is orthonormal. (b) Using inner products, compute [q]B for q(x) = 1+x+x².

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.3: Orthonormal Bases:gram-schmidt Process
Problem 71E
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2. Consider the inner product space (P2(-1, 1), (,)) with inner product (f,9) f(x)g(x) dx, Vf, 9 € P2(-1, 1). (a) Consider the following basis B = {Po: P1, P2} for P2(-1, 1) with Po(x) = 1, P₁(x) = x, P2(x) = (3x² - 1). Verify that B is indeed an orthogonal set. Also, compute the norm of each element of B and conclude whether or not the set is orthonormal. (b) Using inner products, compute [q]B for q(x) = 1+x+x².
Transcribed Image Text:2. Consider the inner product space (P2(-1, 1), (,)) with inner product (f,9) f(x)g(x) dx, Vf, 9 € P2(-1, 1). (a) Consider the following basis B = {Po: P1, P2} for P2(-1, 1) with Po(x) = 1, P₁(x) = x, P2(x) = (3x² - 1). Verify that B is indeed an orthogonal set. Also, compute the norm of each element of B and conclude whether or not the set is orthonormal. (b) Using inner products, compute [q]B for q(x) = 1+x+x².
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