1. Consider R3 with the standard inner product where u = 3 = Σxili i=1 (x1, x2, x3) and v = (31, 32, 3) are vectors in R3. Let U = {(x, y, z) = R³ | 2x-3y+z = 0} (a) Find an orthonormal basis of U. (b) Find the best approximation of v = (1, 1, 1) to the subspace U. 2. Consider P2(R) with the inner product = LP 2 p(x)q(x)dx Use Gram-Schmidt process to transform the basis {1, x, x²) to an orthomormal basis of P2(R).

1. Consider R3 with the standard inner product where u = 3 = Σxili i=1 (x1, x2, x3) and v = (31, 32, 3) are vectors in R3. Let U = {(x, y, z) = R³ | 2x-3y+z = 0} (a) Find an orthonormal basis of U. (b) Find the best approximation of v = (1, 1, 1) to the subspace U. 2. Consider P2(R) with the inner product = LP 2 p(x)q(x)dx Use Gram-Schmidt process to transform the basis {1, x, x²) to an orthomormal basis of P2(R).

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 41E: Use the inner product u,v=2u1v1+u2v2 in R2 and Gram-Schmidt orthonormalization process to transform...

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