MA212-FinalExam-Answers

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

and

, then which of the following cannot exist?

Choose one answer.

	A.
	B.
	C.
	D.

Question 3

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

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

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

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

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

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

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

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

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.