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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.CM: Cumulative Review

Problem 16CM

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Apply the Gram-Schmidt orthonormalization process to transform the given basis for a subspace of R" into an orthonormal
basis for the subspace. Use the vectors in the order in which they are given.
B = {(-5, -3, 8)}

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Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector space?
Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.
Use the inner product u,v=2u1v1+u2v2 in R2 and Gram-Schmidt orthonormalization process to transform {(2,1),(2,10)} into an orthonormal basis.
Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
Find the projection of the vector v=[102]T onto the subspace S=span{[011],[011]}.
Apply the Gram-Schmidt orthonormalization process to transform the given basis for a subspace of Rn into an orthonormal basis for the subspace. Use the vectors in the order in which they are given.B = {(1, 3, 0), (3, 0, −3)}
Apply the Gram-Schmidt orthonormalization process to transform the given basis for a subspace of R^3 into an orthonormal basis for the subspace. Basis: B={(1,3,0), (3,0,-3)}
Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R3 spanned by v1 = (1, 1, 0) and v2 = (2, 1, 1)
Use the Gram-Schmidt process to construct an orthonormal basis for the subspace W of R4 spanned by (1, 1, 0, 0),(2, -1, 0, 1),(3, -3, 0, -2),(1, -2, 0, -3)
Apply Gram-Schmidt orthonormalization process to transform the given basis for a subspace of Rn into an orthonormal basis for a subspace. Use the vectors in the order in which they are given. p = {(3, 4, 0), (2, 0, 0)}

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