1

Suppose that matrix is obtained by performing a sequence of row operations on . Can be obtained by performing row operations on ?

Choose one answer.

A. No | ||

B. In special cases | ||

C. Yes | ||

D. There is not enough information provided to determine this. |

Question 2

If and , then which of the following cannot exist?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 3

If and are symmetric matrices, then the product is also symmetric only when and are

Choose one answer.

A. matrices that commute. | ||

B. square matrices. | ||

C. Hermitian matrices. | ||

D. invertible matrices. |

Question 4

Which of the following expresses the complex number in polar coordinates?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 5

Let be a field. Then, is an ordered field if there exists an order that satisfies which of the following properties?

Choose one answer.

A. For any either or . | ||

B. If and either or , then . | ||

C. If , , then . | ||

D. All of these. |

Question 6

has which of the following properties?

Choose one answer.

A. Boundedness | ||

B. Finiteness | ||

C. Archimedean | ||

D. All of these |

Question 7

For , which of the following inequalities is true?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. All of these |

Question 8

Two matrices are row equivalent if and only if

Choose one answer.

A. one can be obtained from the other by a finite number of elementary row operations. | ||

B. one is the negative of the other. | ||

C. the product of the two matrices is zero. | ||

D. none of these. |

Question 9

Which matrix below would you row reduce to solve the following system of equations?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 10

The complex numbers are a field in part because

Choose one answer.

A. for every non-zero there is a such that . | ||

B. . | ||

C. is a real vector space. | ||

D. for all . |

Question 11

Which of the following fields has the Archimedean property?

Choose one answer.

A. | ||

B. Every finite field | ||

C. The field of rational functions with real coefficients | ||

D. None of these |

Question 12

Determine the angle between the vectors and in .

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 13

Which of the following computes the length of the vector ?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 14

According to the fundamental theorem of algebra, how many complex solutions does the equation have, counted with multiplicity?

Choose one answer.

A. Eight | ||

B. At least one | ||

C. Exactly two | ||

D. Infinitely many |

Question 15

Fill in the blanks. Consider the polynomial . Counted with multiplicity, this polynomial has ________ roots in and ________ roots in .

Choose one answer.

A. at least one; exactly five | ||

B. zero; exactly five | ||

C. exactly five; exactly five | ||

D. zero; exactly five |

Question 16

Calculate the determinant of .

Choose one answer.

A. 24 | ||

B. 10 | ||

C. 12 | ||

D. 42 |

Question 17

Find the eigenvalues of .

Choose one answer.

A. , | ||

B. , | ||

C. , | ||

D. , |

Question 18

What are the eigenvalues of ?

Choose one answer.

A. Negative | ||

B. Real | ||

C. Complex | ||

D. All of these |

Question 19

Calculate the eigenvalues of .

Choose one answer.

A. -1, degenerate | ||

B. 1, -1 | ||

C. 1 degenerate | ||

D. 0, -1 |

Question 20

17. If , then compute and determine whether or not is a normal matrix.

Choose one answer.

A. , is normal. | ||

B. , is not normal. | ||

C. , is normal. | ||

D. , is not normal. |

Question 21

The eigenvalues of a real skew-symmetric matrix must be

Choose one answer.

A. zero. | ||

B. non-negative. | ||

C. purely real. | ||

D. purely imaginary. |

Question 22

Determine the Jordan form for the matrix .

Choose one answer.

A. | ||

B. | ||

C. | ||

D. None of these |

Question 23

Compute the Jordan canonical form for the matrix .

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 24

The determinant rank of the matrix is , where is the

Choose one answer.

A. smallest number such that some submatrix of has a non zero determinant. | ||

B. largest number such that some submatrix of has a non zero determinant. | ||

C. largest number such that some submatrix of has a nonzero determinant. | ||

D. none of these. |

Question 25

Suppose is the zero map defined by for all . Then what is an eigenvector of ?

Choose one answer.

A. | ||

B. | ||

C. Every nonzero vector is an eigenvector of | ||

D. All of these. |

Question 26

What does Schurâ€™s Theorem say?

Choose one answer.

A. Every square matrix is unitarily similar to an upper triangular matrix. | ||

B. Every matrix is unitarily similar to a square matrix. | ||

C. Every matrix is unitarily similar to a triangular matrix. | ||

D. Every upper triangular matrix is similar to a square matrix. |

Question 27

For the set , which of the following properties is not true?

Choose one answer.

A. Composition of permutations is associative. | ||

B. Composition of permutations is commutative. | ||

C. There is an identity element for composition. | ||

D. There is an inverse element for composition. |

Question 28

Fill in the blanks. Let be an matrix. Then equals __________ and equals _________.

Choose one answer.

A. sum of the eigenvalues of ; negative of the sum of the eigenvalues of | ||

B. product of the eigenvalues of ; sum of the eigenvalues of | ||

C. sum of the eigenvalues of ; product of the eigenvalues of | ||

D. ; |

Question 29

What does the second derivative test tell us?

Choose one answer.

A. Whether or not critical points of a function exist | ||

B. Where the derivative is zero for certain types of functions | ||

C. Whether a critical point of a function is a local minimum or a maximum | ||

D. All of these |

Question 30

Suppose is a finite dimensional vector space and is a linear operator on . Then which of the following conditions must be true for a subspace to be an invariant subspace under ?

Choose one answer.

A. for all | ||

B. for all | ||

C. for all | ||

D. for all |

Question 31

Suppose is a linear operator. Then is an eigenvalue of if and only if

Choose one answer.

A. is not injective. | ||

B. is not injective. | ||

C. is invertible. | ||

D. is surjective. |

Question 32

Let be a finite dimensional vector space and let be invertible. Then is an eigenvalue for if and only if

Choose one answer.

A. is an eigenvalue for . | ||

B. is an eigenvalue for . | ||

C. is an eigenvalue for . | ||

D. all of these. |

Question 33

If is a real matrix and one of its eigenvalues is with eigenvector , then what can be said about another one of its eigenvalues and eigenvectors?

Choose one answer.

A. and is the conjugate of | ||

B. and is the conjugate of | ||

C. and is the negative conjugate of | ||

D. None of these. |

Question 34

Suppose one eigenvalue of a real matrix is and the corresponding eigenvector is . Then which of the following must also be an eigenpair of ?

Choose one answer.

A. and | ||

B. and | ||

C. and | ||

D. and |

Question 35

Suppose that is an eigenpair of the invertible matrix . Then is an eigenpair of which of the following matrices?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 36

A quadratic form in three dimensions can be written as , where satisfies which of the following properties?

Choose one answer.

A. is a symmetric matrix. | ||

B. is a orthogonal matrix. | ||

C. is a normal matrix. | ||

D. is a unitary matrix. |

Question 37

Suppose at a critical point of a function , the Hessian matrix is given by . Then what does the second derivative test tell us about this critical point?

Choose one answer.

A. It is a local maximum. | ||

B. It is a local minimum. | ||

C. It is a saddle point. | ||

D. It is zero. |

Question 38

Let where is an -dimensional -vector space. Then is guaranteed to have an eigenvalue when

Choose one answer.

A. the minimal polynomial for has a root in . | ||

B. the minimal polynomial for has no roots in . | ||

C. is an isomorphism. | ||

D. the only -invariant subspaces of are trivial. |

Question 39

Let . Determine the Jordan normal form for .

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 40

What are the proper subspaces of ?

Choose one answer.

A. Trivial subspaces | ||

B. Lines through the origin | ||

C. Both A and B | ||

D. None of these |

Question 41

Let be subspaces, then , if and only if two conditions hold. One is that . What is the other condition?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 42

Which of the following subsets of is NOT a subspace of ?

Choose one answer.

A. Symmetric matrices | ||

B. Diagonal matrices | ||

C. Nonsingular matrices | ||

D. Upper triangular matrices |

Question 43

Fill in the blank. The ___________ of orthogonal vectors is -dimensional.

Choose one answer.

A. collection | ||

B. span | ||

C. kernel | ||

D. transformation |

Question 44

Which of the following is NOT a vector space associated with the matrix ?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 45

Suppose is a nonsingular matrix. Then which of the following best describes its four fundamental subspaces?

Choose one answer.

A. and | ||

B. and | ||

C. and | ||

D. |

Question 46

If and are subspaces of a vector space , then equals which of the following?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 47

Consider the following linear transformation where rotates each vector 90 degrees about the -axis. Find the matrix representation of in the standard basis.

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 48

Which of the following is a basis of ?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. All of these |

Question 49

Let be a complex vector space. Let , and . Then, which of the following is true?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. None of these |

Question 50

Fill in the blank. A linear map is _____________, if and only if is injective and surjective.

Choose one answer.

A. singular | ||

B. invertible | ||

C. continuous | ||

D. one-to-one |

Question 51

Let be such that for all written in the standard basis. Then, is a

Choose one answer.

A. zero map. | ||

B. identity map. | ||

C. vector space. | ||

D. surjective linear map. |

Question 52

Consider the following linear transformation : â†’ where rotates each vector 90 degrees counterclockwise about the -axis and then rotates 45 degrees counterclockwise about the -axis. Find the matrix representation of in the standard basis.

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 53

Consider the following linear transformation : â†’ , where rotates each vector 90 degrees counterclockwise about the -axis and then rotates 45 degrees counterclockwise about the -axis. Find in the standard basis.

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 54

What is the characteristic polynomial of the matrix ?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 55

Let and let be defined by . Then what is the matrix, , for using the standard basis?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. None of these |

Question 56

All have a singular value decomposition, meaning there exists an orthonormal basis and such that , where the 's are

Choose one answer.

A. any positive numbers. | ||

B. singular values of . | ||

C. complex values. | ||

D. obtained from the dot product of and . |

Question 57

An matrix is called a Markov matrix if the following is satisfied:

Choose one answer.

A. for all and | ||

B. for all and | ||

C. for all and | ||

D. for all and |

Question 58

What can be said about eigenvalues of a Markov matrix?

Choose one answer.

A. Every eigenvalue of a Markov matrix satisfies . | ||

B. Every eigenvalue of a Markov matrix satisfies . | ||

C. Every eigenvalue of a Markov matrix satisfies . | ||

D. All of these. |

Question 59

What does it mean for a matrix to be stochastic?

Choose one answer.

A. has negative entries and . | ||

B. has non-negative entries and . | ||

C. has non-negative entries and . | ||

D. |

Question 60

If is the union of the coordinate axes, then is NOT a real vector space because

Choose one answer.

A. it is not closed under addition. | ||

B. it is infinite. | ||

C. it is not closed under multiplication. | ||

D. it does not contain a zero vector. |

Question 61

Suppose , , and are invertible matrices. Which of the following is false?

Choose one answer.

A. does not have full rank. | ||

B. | ||

C. is invertible. | ||

D. |

Question 62

Fill in the blank. Columns of an matrix are an orthonormal basis for , if and only if is a _____________ matrix.

Choose one answer.

A. normal | ||

B. unitary | ||

C. symmetric | ||

D. square |

Question 63

Using the standard inner product, which of the following pairs are orthogonal vectors in ^{? }

Choose one answer.

A. , | ||

B. , | ||

C. , | ||

D. None of these |

Question 64

The spectral theorem states which of the following?

Choose one answer.

A. Normal operators are diagonal with respect to an orthonormal basis. | ||

B. Normal operators are diagonal with respect to a singular set of vectors. | ||

C. Nilpotent operators are diagonal with respect to an orthonormal basis. | ||

D. Semi-symmetric operators are diagonal with respect to any basis. |

Question 65

Given , the adjoint of is defined to be the operator , such that which is true?

Choose one answer.

A. for all . | ||

B. for all . | ||

C. for all . | ||

D. None of these |

Question 66

Which of the following best describes every eigenvalue of a self-adjoint operator?

Choose one answer.

A. Real | ||

B. Complex | ||

C. Degenerate | ||

D. All of these |

Question 67

According to the spectral theorem, if is a finite dimensional inner product space over and , then which of the following must be true?

Choose one answer.

A. is normal if and only if there exists an orthonormal basis for consisting of eigenvectors for . | ||

B. is self-adjoint if and only if the eigenvectors of are degenerate. | ||

C. is normal if and only if is diagonalizable. | ||

D. None of these. |

Question 68

What are operators that preserve the inner product called?

Choose one answer.

A. Orthogonal | ||

B. Isometries | ||

C. Normal | ||

D. All of these |

Question 69

Which of the following is an example of a normal operator?

Choose one answer.

A. Symmetric matrices | ||

B. Hermitian matrices | ||

C. Orthogonal matrices | ||

D. All of these |

Question 70

In order to unitarily diagonalize an matrix , what do you need to do?

Choose one answer.

A. Find a unitary matrix and a diagonal matrix such that . | ||

B. Find a unitary matrix and a diagonal matrix such that . | ||

C. Find a unitary matrix and a unitary matrix such that . | ||

D. Find a diagonal matrix and a unitary matrix such that . |

Question 71

It is possible for an operator to be normal but not satisfy which of the following conditions?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. All of these |

Question 72

For all , is

Choose one answer.

A. self-adjoint. | ||

B. nilpotent. | ||

C. orthogonal. | ||

D. diagonal. |

Question 73

Let and let be a normal operator such that . Then, which of the following is true?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. All of these |

Question 74

If is a complex vector space and is a normal operator with only one distinct eigenvalue , then which of the following is true?

Choose one answer.

A. | ||

B. for every | ||

C. for every | ||

D. All of these |

Question 75

Suppose and are two distinct eigenvalues of a real symmetric matrix . Then what are their corresponding eigenvectors?

Choose one answer.

A. Similar | ||

B. Equal | ||

C. Orthogonal | ||

D. None of these |

Question 76

Which of the following properties should a norm on a normed linear space satisfy?

Choose one answer.

A. for all , and if and only if | ||

B. for all | ||

C. for all $$v,w \in V$ | ||

D. All of these |

Question 77

According to the Cauchy-Schwarz inequality, on any inner product space which of the following must be true?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 78

Fill in the blanks. Suppose and are inner product spaces, , and . Then the tensor product is an element of ______ defined by _______.

Choose one answer.

A. ; | ||

B. ; | ||

C. ; | ||

D. ; |

Question 79

What is the Riesz representation theorem?

Choose one answer.

A. Let where is an inner product space. Then there exists a unique such that for all , . | ||

B. Let where is an inner product space. Then there exists a unique such that for all , . | ||

C. Let where is an inner product space. Then there exist infinitely many such that for all , . | ||

D. None of these. |

Question 80

A Hilbert space is

Choose one answer.

A. a complete inner product space. | ||

B. an inner product space. | ||

C. a normed space. | ||

D. none of these. |

Question 81

In an abstract vector space, what does the norm measure?

Choose one answer.

A. The angle between two vectors | ||

B. The length of a vector | ||

C. The direction of a vector | ||

D. All of these |

Question 82

Consider the vector space with basis vectors and . Applying the Gram-Schmidt process to produces which orthonormal basis for ?

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 83

If , , and are three linearly independent vectors in , then the volume of the parallelepiped determined by , , and is

Choose one answer.

A. . | ||

B. . | ||

C. . | ||

D. . |

Question 84

Let be an matrix with characteristic polynomial . Which of the following is true?

Choose one answer.

A. | ||

B. is invertible. | ||

C. | ||

D. |

Question 85

Let , , and . Which of the following is true?

Choose one answer.

A. is not orthogonal to either or . | ||

B. and are orthogonal. | ||

C. and are orthogonal. | ||

D. and are orthogonal. |

Question 86

Let . Determine the range-nullspace decomposition of relative to .

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 87

Let . If is the singular-value decomposition of , then

Choose one answer.

A. | ||

B. | ||

C. | ||

D. |

Question 88

Let be a linear transformation. Suppose the matrix for relative to a basis for is . Suppose is the transition matrix from another basis to . Then the matrix for with respect to is

Choose one answer.

A. . | ||

B. . | ||

C. . | ||

D. . |

Question 89

Let be an inner product space with inner product . Then the length of a vector is

Choose one answer.

A. . | ||

B. . | ||

C. . | ||

D. . |