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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.CR: Review Exercises

Problem 59CR

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Determine if the set of vectors is a basis of R*. If not, determine the dimension of the subspace spanned by the vectors.
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The dimension of the subspace spanned by the vectors is

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Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?
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.
Proof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.
Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). 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 orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are nonnegative.
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are Pythagorean triples.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose second component is the square of the first.
Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all matrices in Mn,n with determinants equal to 1.
Subsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose components are integers.
Subsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose components are rational numbers.

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