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 Schur’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 : 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 : , 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. .
.
.