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Exam 8: Orthogonality

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Exam 1: Systems of Linear Equations57 Questionsadd course
Exam 2: Euclidean Space48 Questionsadd course
Exam 3: Matrices76 Questionsadd course
Exam 4: Subspaces60 Questionsadd course
Exam 5: Determinants48 Questionsadd course
Exam 6: Eigenvalues and Eigenvectors75 Questionsadd course
Exam 7: Vector Spaces45 Questionsadd course
Exam 8: Orthogonality75 Questionsadd course
Exam 9: Linear Transformations60 Questionsadd course
Exam 10: Inner Product Spaces45 Questionsadd course
Exam 11: Additional Topics and Applications75 Questionsadd course
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Find an equation for the plane in Find an equation for the plane in that best fits the given data. , , , that best fits the given data. Find an equation for the plane in that best fits the given data. , , , , Find an equation for the plane in that best fits the given data. , , , , Find an equation for the plane in that best fits the given data. , , , , Find an equation for the plane in that best fits the given data. , , ,

(Essay)
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(87)
Question 21

If If and are nonzero subspaces of , then for every vector u in . and If and are nonzero subspaces of , then for every vector u in . are nonzero subspaces of If and are nonzero subspaces of , then for every vector u in . , then If and are nonzero subspaces of , then for every vector u in . for every vector u in If and are nonzero subspaces of , then for every vector u in . .

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Question 22

Find the QR decomposition for the matrix Find the QR decomposition for the matrix . .

(Essay)
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Question 23

Find a singular value decomposition for the matrix Find a singular value decomposition for the matrix . .

(Essay)
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Question 24

Determine the numerical rank of a Determine the numerical rank of a matrix with singular values , , , and , if . matrix with singular values Determine the numerical rank of a matrix with singular values , , , and , if . , Determine the numerical rank of a matrix with singular values , , , and , if . , Determine the numerical rank of a matrix with singular values , , , and , if . , and Determine the numerical rank of a matrix with singular values , , , and , if . , if Determine the numerical rank of a matrix with singular values , , , and , if . .

(Short Answer)
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Question 25

Determine Determine , where , . , where Determine , where , . , Determine , where , . .

(Essay)
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Question 26

Find a basis for Find a basis for for the subspace S. for the subspace S. Find a basis for for the subspace S.

(Essay)
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Question 27

Find all values of a (if any) so that the given vectors form an orthogonal set. Find all values of a (if any) so that the given vectors form an orthogonal set.

(Essay)
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Question 28

Find an orthonormal basis for the subspace Find an orthonormal basis for the subspace , and use it to to find , where . , and use it to to find Find an orthonormal basis for the subspace , and use it to to find , where . , where Find an orthonormal basis for the subspace , and use it to to find , where . .

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Question 29

If A is symmetric, and If A is symmetric, and and with , then is orthogonal to . and If A is symmetric, and and with , then is orthogonal to . with If A is symmetric, and and with , then is orthogonal to . , then If A is symmetric, and and with , then is orthogonal to . is orthogonal to If A is symmetric, and and with , then is orthogonal to . .

(True/False)
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Question 30

Find a singular value decomposition for the matrix Find a singular value decomposition for the matrix . .

(Essay)
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Question 31

If S is a subspace of If S is a subspace of , then . , then If S is a subspace of , then . .

(True/False)
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Question 32

Find the least squares solution for the given system. Find the least squares solution for the given system.

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Question 33

If A is any matrix, then If A is any matrix, then is diagonalizable. is diagonalizable.

(True/False)
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Question 34

Find an equation for the line in Find an equation for the line in that best fits the given data. , , that best fits the given data. Find an equation for the line in that best fits the given data. , , , Find an equation for the line in that best fits the given data. , , , Find an equation for the line in that best fits the given data. , ,

(Essay)
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Question 35

Determine the numerical rank of a Determine the numerical rank of a matrix with singular values , , , , and , if . matrix with singular values Determine the numerical rank of a matrix with singular values , , , , and , if . , Determine the numerical rank of a matrix with singular values , , , , and , if . , Determine the numerical rank of a matrix with singular values , , , , and , if . , Determine the numerical rank of a matrix with singular values , , , , and , if . , and Determine the numerical rank of a matrix with singular values , , , , and , if . , if Determine the numerical rank of a matrix with singular values , , , , and , if . .

(Short Answer)
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Question 36

If A is any matrix, then If A is any matrix, then . .

(True/False)
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Question 37

If If is a matrix and is in , then is a least squares solution to . is a matrix and If is a matrix and is in , then is a least squares solution to . is in If is a matrix and is in , then is a least squares solution to . , then If is a matrix and is in , then is a least squares solution to . is a least squares solution to If is a matrix and is in , then is a least squares solution to . .

(True/False)
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Question 38

Apply the Gram-Schmidt process to the given basis for Apply the Gram-Schmidt process to the given basis for to produce an orthogonal basis for . Then normalize the vectors to produce an orthonormal basis for . to produce an orthogonal basis for Apply the Gram-Schmidt process to the given basis for to produce an orthogonal basis for . Then normalize the vectors to produce an orthonormal basis for . . Then normalize the vectors to produce an orthonormal basis for Apply the Gram-Schmidt process to the given basis for to produce an orthogonal basis for . Then normalize the vectors to produce an orthonormal basis for . . Apply the Gram-Schmidt process to the given basis for to produce an orthogonal basis for . Then normalize the vectors to produce an orthonormal basis for .

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Question 39

Every system of equations has at least one least squares solution.

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Question 40
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