If two vectors are orthogonal, then which of the following is true?

Choose one answer.

	a. Their dot product is 0.
	b. Their cross product is 0.
	c. Their dot product is 1.
	d. Both A and B
	e. Both B and C

Question 2

Let

, and let

. What is

Answer:

Question 3

Let

, and let

. If

, what is

Answer:

Question 4

Let

and let

then what is

Answer:

Question 5

Let

, and let

. If

, then what is

Answer:

Question 6

Consider this system:

. What is

Answer:

Question 7

Consider this system:

. What is

Answer:

Question 8

Consider this system:

. What is

Answer:

Question 9

Consider a parallelepiped

with three edges meeting at a vertex at the origin, given these vectors:

, and

. What is its volume?

Answer:

Question 10

What is the angle (in radians) between the vectors

and

Answer:

Question 11

Find the length of the projection of the vector

onto the vector

Answer:

Question 12

is a vector in the plane then the vector

Choose one answer.

	a. makes an angle of with and is twice as long
	b. makes and angle of with and is twice as long
	c. is perpendicular to
	d. makes an angle of with

Question 13

Calculate the determinant of this matrix:

Answer:

Question 14

Suppose that

is a

matrix and that

. Let

be an elementary matrix whose only action is to swap two rows of

. Let

. What is

Answer:

Question 15

Consider matrix

has this format:

. What is the value of

Answer:

Question 16

Suppose

is matrix in

, and

is a vector in

. Evaluate the following statement: there exists solution

to the equation

Choose one answer.

	a. For this statement to be true, it is necessary that be nonsingular.
	b. For this statement to be true, it is sufficient that be nonsingular.
	c. For this statement to be true, it is necessary and sufficient that be nonsingular.
	d. For this statement to be true, it is neither necessary nor sufficient that be nonsingular.
	e. If is nonsingular, then this statement is false.

Question 17

A system of linear equations with six unknowns can have how many solutions?

Choose one answer.

	a. 0
	b. 1
	c. Infinitely many
	d. All of these answers
	e. None of these answers

Question 18

Consider the linear map

, which acts by rotating a vector

degrees counterclockwise around the origin. Which of the following is the matrix of the linear transformation?

Choose one answer.

	a.
	b.
	c.
	d.
	e.

Question 19

Suppose that

and

Consider the linear map

which satisfies

and

Let

what is

Answer:

Question 20

Similar matrices always have which of the following in common?

Choose one answer.

	a. Column space
	b. Determinant
	c. Nullspace
	d. Eigenvectors
	e. Inverse

Question 21

What is the dimension of the span of the set

Answer:

Question 22

Suppose that the columns of an

matrix

are linearly independent. Then, which of the following statements is always true?

Choose one answer.

	a. The kernel of is 0.
	b. The determinant of is 1.
	c. has linearly independent eigenvectors.
	d. is idempotent.
	e. is a projection matrix.

Question 23

Let

be a

matrix. Let

be a vector in the nullspace of

. Let

be another

matrix. Evaluate the following statement:

Choose one answer.

	a. In order for this statement to be true, it is necessary that be a multiple of the identity.
	b. In order for this statement to be true, it is sufficient that be a multiple of the identity.
	c. In order for this statement to be true, it is necessary and sufficient that be a multiple of the identity.
	d. In order for this statement to be true, it is neither necessary nor sufficient that be a multiple of the identity.
	e. This statement is false if is a multiple of the identity.

Question 24

Suppose that

is a

matrix with determinant 2. What is

Answer:

Question 25

Consider these maps from

.

Which is linear?

Choose one answer.

	a.
	b.
	c.
	d. All of these answers
	e. None of these answers

Question 26

Two matrices are row-equivalent if and only if which of the following is true?

Choose one answer.

	a. They have the same nullspace.
	b. They have the same eigenspace.
	c. One can be changed to the other by a sequence of elementary row operations.
	d. Both A and C
	e. Both B and C

Question 27

Suppose a

matrix

is row-equivalent to the identity matrix. Which of the following may be false?

Choose one answer.

	a. .
	b. The rowspace of is .
	c. The eigenvalues of are nonzero.
	d. The nullspace of is the zero vector.
	e. is nonsingular.

Question 28

Consider the matrix

what is

Answer:

Question 29

True or False: If A is a matrix and b is not the zero vector, then Ax = b has a solution if and only if b can be written as a linear combination of the columns of A.

Answer:

True False

Question 30

True or False: These vectors are linearly independent:

Answer:

True False

Question 31

Calculate the determinant of this matrix:

Answer:

Question 32

True or False: If A is a 12 x 15 matrix, the null-space of A must have dimension at least 1?

Answer:

True False

Question 33

Suppose that

is a system of equations with

. The solution vector is

Answer:

True False

Question 34

If the rank of a 4 x 4 matrix is 3 then its determinant

Choose one answer.

	a. =1
	b. =0
	c. cannot be determined

Question 35

If a 4 x 4 matrix is in row-echelon form the the entry in the third row, second column must be 0.

Answer:

True False

Question 36

True or False: A set of three non-zero vectors in the plane can be linearly independent.

Answer:

True False

Question 37

True or false: every non-vertical straight line in the plane is a subspace of

Answer:

True False

Question 38

is an n x n matrix of rank

, then what is the dimension of the null-space of

Choose one answer.

	a. p
	b. n-p
	c. n-p+1
	d. 0
	e. Cannot be determined from the data

Question 39

True or False: To find an eigenvector of a matrix

corresponding to an eigenvalue

, simply find any vector in the nullspace of

Answer:

True False

Question 40

Consider this matrix:

Clearly,

is an eigenvalue of

. Let

be the eigenvector corresponding to

. If

, what is

Answer:

Question 41

Consider this matrix:

Clearly,

is an eigenvalue of

. Let

be the eigenvector corresponding to

. If

, what is

Answer:

Question 42

Consider this matrix:

The characteristic polynomial of

is of the form

What is

Answer:

Question 43

Suppose matrix

is an

matrix with characteristic polynomial

. What is the largest possible number of linearly independent solutions to

Answer:

Question 44

Suppose a matrix

is a m x m matrix, which has characteristic polynomial

. Is

diagonalizable? Why, or why not?

Choose one answer.

	a. No, because is singular.
	b. No, because has repeated roots.
	c. Maybe, it depends on whether has a degree equal to the dimension of the space.
	d. Maybe, it depends on the dimensions of the eigenspaces.
	e. Yes, because all the roots of are real.

Question 45

Suppose a

matrix

is diagonalizable. Then,

always has which of the following?

Choose one answer.

	a. distinct eigenvalues
	b. linearly independent eigenvectors
	c. Exactly as many linearly independent eigenvectors as eigenvalues
	d. nonzero eigenvalues
	e. linearly independent columns

Question 46

Consider this matrix:

What is the largest eigenvalue of

Answer:

Question 47

Compute the smallest (i.e., most negative) eigenvalue of this matrix:

Answer:

Question 48

Suppose that

is an eigenvector of a matrix

corresponding to a nonzero eigenvalue

. True or False:

is always solvable.

Answer:

True False

Question 49

True or False: If two matrices have the same characteristic polynomial, then they are similar.

Answer:

True False

Question 50

True or false: The polynomial

is the characteristic polynomial of a symmetric matrix.

Answer:

True False

Question 51

Let

and

is the orthonormal basis of

obtained by the Gram-Schmidt process with

then

Choose one answer.

	a. 1
	b.
	c. 0
	d. -1
	e.

Question 52

True or false: the matrix

is orthogonal?

Answer:

True False

Question 53

is an

orthogonal matrix, which of the following may not be true about

Choose one answer.

	a. The determinant of equals
	b.
	c. The rows of form an orthonormal basis for .
	d. The columns of form an orthonormal basis for .
	e. The nullity of is

Question 54

The line which gives the best least squares approximation to the points in the plane with coordinates

has slope

Choose one answer.

	a. 3/2
	b. 2/3
	c. -1/3
	d. 0
	e. 2

Question 55

Suppose A is a 3 x 3 symmetric matrix,

is an eigenvalue of algebraic multiplicity 2 and

is an eigenvalue of algebraic multiplicity 1. If

is an eigenvector corresponding to

and

is an eigenvector corresponding to

, then x=

Answer:

Question 56

What is the geometric multiplicity of the eigenvalus

of the matrix

Answer:

Question 57

What is the geometric multiplicity of the eigenvalue

of the matrix

Answer:

Question 58

is a 4 x 4 real matrix then the matrix

Choose one answer.

	a. is always symmetric
	b. is always skew-symmetric
	c. is always singular
	d. is always non-singular

Question 59

is a 4 x 4 real matrix which is true of the matrix

1. must have non-negative real eigenvalues
2. must have purely imaginary eigenvalues
3. must have an orthonormal basis of eigenvectors
4. is diagonalizable

Choose one answer.

	a. 1., 3. and 4.
	b. 2. and 3.
	c. 2., 3. and 4
	d. 1. and 2.
	e. 1., 2., and 3.

Question 60

True or False: In a vector space of dimension

, any spanning set of cardinality

is a basis.

Answer:

True False

Question 61

Which of the following is required of a set

with a well-defined addition of elements and scalar multiplication over a field

to be a vector space?

Choose one answer.

	a. Every element of has an additive inverse.
	b. If and and , then
	c. has a well-defined inner product.
	d. All of the above
	e. A and B only

Question 62

Consider

, which is the vector space of functions that are continuous on the interval

. Let

.

True or False:

is a linear subspace of

Answer:

True False

Question 63

Consider a

matrix

with eigenvalues

and

, each having multiplicity

True or False: The set of all eigenvectors of

form a subspace of

Answer:

True False

Question 64

Consider a

matrix

with eigenvalues -1, 4, and 5.

True or False: The set of all right eigenvectors of

, which correspond to the eigenvector 4, form a subspace of

Answer:

True False

Question 65

True or False: Trace is a linear operator on the vector space of

real-valued matrices endowed with the usual matrix addition and scalar multiplication.

(Recall that for a matrix

Answer:

True False

Question 66

Suppose that

and

are bases for

. Suppose that

and

are orthogonal vectors in

with coordinates corresponding to basis

. Let

and

be the coordinate vectors for

and

in the basis

.

True or False:

is orthogonal to

Answer:

True False

Question 67

For any given real

matrix

, evaluate the following statement:

Choose one answer.

	a. For this statement to be true, it is necessary that .
	b. For this statement to be true, it is sufficient that .
	c. For this statement to be true, it is necessary and sufficient that .
	d. For this statement to be true, it is neither necessary nor sufficient that .
	e. This statement is false if .