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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Question
![1. Consider R3 with the standard inner product
where u =
3
<u,v>= Σxili
i=1
(x1, x2, x3) and v = (31, 32, y3) 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
<p,q>=
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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6842f34-ce08-42da-92c9-60453cd22454%2F64a190b0-1f2b-423d-9c37-16f90ad3016f%2F0a6buy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Consider R3 with the standard inner product
where u =
3
<u,v>= Σxili
i=1
(x1, x2, x3) and v = (31, 32, y3) 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
<p,q>=
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).
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