Question 1

If S is a nonzero subspace of  , then for every vector u in  ,  belongs to S.

Accepted Answer

A) True 
 B)False  True

Question 2

If  is a nonzero subspace of  , and u belongs to  , then  for every vector  .

Accepted Answer

A) True 
 B)False  False

Question 3

If  is an invertible (square) matrix with singular value decomposition  , then a singular value decomposition for  is given by  .

Accepted Answer

A) True 
 B)False  False

Question 4

Find the QR decomposition for the matrix  .

Accepted Answer

The answer of Find the QR decomposition for the matrix...

Question 5

If  is a QR factorization of a matrix A, then  .

Accepted Answer

A) True 
 B)False

Question 6

Find an orthogonal basis for the subspace  .

Accepted Answer

The answer of Find an orthogonal basis for the subspace...

Question 7

Suppose  and  are orthogonal with  and  . Determine  .

Accepted Answer

The answer of Suppose  and  are orthogonal with...

Question 8

Find the singular values for the matrix  .

Accepted Answer

The answer of Find the singular values for the matrix...

Question 9

If Q is an orthogonal matrix, then  .

Accepted Answer

A) True 
 B)False

Question 10

If  is an orthogonal set of vectors, and  , then
  .

Accepted Answer

A) True 
 B)False

Question 11

If a system of equations has more equations than variables, then the system has a unique least squares solution.

Accepted Answer

A) True 
 B)False

Question 12

Find the normal equations for the given system.

Accepted Answer

The answer of Find the normal equations for the given...

Question 13

Determine if the vectors form an orthogonal set.

Accepted Answer

The answer of Determine if the vectors form an orthogonal...

Question 14

Determine  from the given matrix with orthogonal columns without using row operations.

Accepted Answer

The answer of Determine  from the given matrix with...

Question 15

Find a singular value decomposition for the matrix  .

Accepted Answer

The answer of Find a singular value decomposition for the...

Question 16

The eigenvalues and corresponding eigenvectors for a symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Accepted Answer

The answer of The eigenvalues and corresponding eigenvectors for a...

Question 17

Find an orthonormal basis for the subspace  .

Accepted Answer

The answer of Find an orthonormal basis for the subspace...

Question 18

Find a basis for  for the subspace S.

Accepted Answer

The answer of Find a basis for  for the...

Question 19

If A is an  diagonalizable matrix, then there exists a diagonal matrix D and an orthogonal matrix P such that  .

Accepted Answer

A) True 
 B)False

Question 20

Verify that the eigenvalues of  are nonnegative.

Accepted Answer

The answer of Verify that the eigenvalues of  are...

Exam 8: Orthogonality

If S is a nonzero subspace of , then for every vector u in , belongs to S.

If is a nonzero subspace of , and u belongs to , then for every vector .

If is an invertible (square) matrix with singular value decomposition , then a singular value decomposition for is given by .

Find the QR decomposition for the matrix .

If is a QR factorization of a matrix A, then .

Find an orthogonal basis for the subspace .

Suppose and are orthogonal with and . Determine .

Find the singular values for the matrix .

If Q is an orthogonal matrix, then .

If is an orthogonal set of vectors, and , then .

If a system of equations has more equations than variables, then the system has a unique least squares solution.

Find the normal equations for the given system.

Determine if the vectors form an orthogonal set.

Determine from the given matrix with orthogonal columns without using row operations.

Find a singular value decomposition for the matrix .

The eigenvalues and corresponding eigenvectors for a symmetric matrix A are given. Find matrices D and P of an orthogonal diagonalization of A.

Find an orthonormal basis for the subspace .

Find a basis for for the subspace S.

If A is an diagonalizable matrix, then there exists a diagonal matrix D and an orthogonal matrix P such that .

Verify that the eigenvalues of are nonnegative.