Deck 8: Orthogonality

Question 1

Find all values of a so that u and v are orthogonal.

Answer 1

Question 2

Find all values of a so that u and v are orthogonal.

Answer 2

Question 3

Determine if the vectors form an orthogonal set.

Answer 3

The vectors form an orthogonal set.

Question 4

Determine if the vectors form an orthogonal set.

Question 5

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

Question 6

Suppose and are orthogonal with and . Determine .

Question 7

Determine all values of a so that the vector u is orthogonal to the subspace S.

Question 8

Find a basis for for the subspace S.

Question 9

Find a basis for for the subspace S.

Question 10

Suppose S is a subspace of , and . Determine .

Question 11

If is an orthogonal basis for a subspace S of , and then is a diagonal matrix.

Question 12

If is a linearly independent set of vectors, then is orthogonal.

Question 13

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

Question 14

If A is any matrix, then .

Question 15

If S is a subspace of , then .

Question 16

Determine , where .

Question 17

Determine , where .

Question 18

Determine , where , .

Question 19

Find an orthogonal basis for the subspace .

Question 20

Find an orthogonal basis for the subspace .

Question 21

Find an orthogonal basis for the subspace .

Question 22

Find , where .

Question 23

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 .

Question 24

Find an orthonormal basis for the subspace .

Question 25

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

Question 26

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

Question 27

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

Question 28

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

Question 29

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

Question 30

If is an orthonormal set, then is linearly independent.

Question 31

Determine if the given matrix is symmetric.

Question 32

Determine if the given matrix is orthogonal.

Question 33

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

Question 34

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

Question 35

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

Question 36

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

Question 37

Verify that the eigenvalues of are nonnegative.

Question 38

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

Question 39

Find the QR decomposition for the matrix .

Question 40

Find the QR decomposition for the matrix .

Question 41

If A is any matrix, then is diagonalizable.

Question 42

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

Question 43

If Q is an orthogonal matrix, then .

Question 44

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

Question 45

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

Question 46

Find the singular values for the matrix .

Question 47

Find the singular values for the matrix .

Question 48

Find a singular value decomposition for the matrix .

Question 49

Find a singular value decomposition for the matrix .

Question 50

Find a singular value decomposition for the matrix .

Question 51

Find a singular value decomposition for the matrix .

Question 52

Express the given matrix as an outer product expansion , where and are the singular values of .

Question 53

Express the given matrix as an outer product expansion , where and are the singular values of .

Question 54

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

Question 55

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

Question 56

Every matrix A has a singular value decomposition.

Question 57

The singular value decomposition of a matrix A is unique.

Question 58

If D is a diagonal matrix with diagonal entries , then the singular values of D are given by .

Question 59

If is a singular value decomposition for a matrix , then is an orthogonal diagonalizing matrix for .

Question 60

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

Question 61

Find the vector in the subspace S that is closest to y.

Question 62

Find the vector in the subspace S that is closest to y.

Question 63

Find the normal equations for the given system.

Question 64

Find the normal equations for the given system.

Question 65

Find the least squares solution for the given system.

Question 66

Find the least squares solution for the given system.

Question 67

Find the least squares solution for the given system.

Question 68

Find all least squares solutions for the given system.

Question 69

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

Question 70

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

Question 71

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

Question 72

If a matrix A has linearly independent columns, then for every vector y there exists a unique least squares solution of .

Question 73

If a system of equations has more variables than equations, then the system has infinitely many least squares solutions.

Question 74

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

Question 75

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