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

	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

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Let , and let . What is ?

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Let , and let . If , what is ?

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Let

and let

If

then what is ?

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Let , and let . If , then what is ?

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Consider this system: . What is ?

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Consider this system: . What is ?

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Consider this system: . What is ?

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Consider a parallelepiped with three edges meeting at a vertex at the origin, given these vectors: , , and . What is its volume?

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What is the angle (in radians) between the vectors



and

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Find the length of the projection of the vector



onto the vector

.

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If is a vector in the plane then the vector

	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

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Calculate the determinant of this matrix: .

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Suppose that is a matrix and that . Let be an elementary matrix whose only action is to swap two rows of . Let . What is ?

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Consider matrix : . has this format: . What is the value of ?

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Suppose is matrix in , and is a vector in . Evaluate the following statement: there exists solution to the equation .

	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.

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

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

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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?

	a.
	b.
	c.
	d.
	e.

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Suppose that


and


Consider the linear map


which satisfies


and


Let


If


what is ?

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Similar matrices always have which of the following in common?

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

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What is the dimension of the span of the set

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Suppose that the columns of an matrix are linearly independent. Then, which of the following statements is always true?

	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.

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Let be a matrix. Let be a vector in the nullspace of . Let be another matrix. Evaluate the following statement: .

	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.

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Suppose that is a matrix with determinant 2. What is ?

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Consider these maps from







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Which is linear?

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

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

	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

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

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

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Consider the matrix


If


what is ?

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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.

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True or False: These vectors are linearly independent:

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Calculate the determinant of this matrix: .

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

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Suppose that

,


is a system of equations with . The solution vector is

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

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

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

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

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

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If is an n x n matrix of rank , then what is the dimension of the null-space of =?

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

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True or False: To find an eigenvector of a matrix corresponding to an eigenvalue of , simply find any vector in the nullspace of .

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Consider this matrix:



Clearly, is an eigenvalue of . Let







be the eigenvector corresponding to . If , what is ?

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Consider this matrix:



Clearly, is an eigenvalue of . Let







be the eigenvector corresponding to . If , what is ?

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Consider this matrix:



The characteristic polynomial of is of the form
What is ?

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Suppose matrix is an matrix with characteristic polynomial . What is the largest possible number of linearly independent solutions to ?

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Suppose a matrix is a m x m matrix, which has characteristic polynomial . Is diagonalizable? Why, or why not?

	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.

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Suppose a matrix is diagonalizable. Then, always has which of the following?

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

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Consider this matrix:



What is the largest eigenvalue of ?

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

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Suppose that is an eigenvector of a matrix corresponding to a nonzero eigenvalue . True or False: is always solvable.

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

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True or false: The polynomial is the characteristic polynomial of a symmetric matrix.

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Let



and



and



If is the orthonormal basis of obtained by the Gram-Schmidt process with






then

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

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True or false: the matrix





is orthogonal?

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If is an orthogonal matrix, which of the following may not be true about .

	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

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

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

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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=

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What is the geometric multiplicity of the eigenvalus of the matrix

?

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What is the geometric multiplicity of the eigenvalue of the matrix

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If is a 4 x 4 real matrix then the matrix

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

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If 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

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

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True or False: In a vector space of dimension , any spanning set of cardinality is a basis.

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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?

	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

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

True or False: is a linear subspace of .

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Consider a matrix with eigenvalues and , each having multiplicity

True or False: The set of all eigenvectors of form a subspace of .

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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 .

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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 , .)

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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 .

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For any given real matrix , evaluate the following statement: .

	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 .

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If denotes the vector space of polynomials of degree at most 4 in one variable and is the linear transformation defined by


,


where denotes the derivative of . If denotes the matrix representing with respect to the basis of , then

	a. The determinant of is .
	b. has rank
	c. The determinant of is .
	d. is symmetric
	e. is orthogonal

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If is the vector space of continuous function on the interval with inner product given by


,


find so that {} is an orthonormal basis for the subspace spanned by {}.

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True or False: The vector space consisting of real numbers over the field of rational numbers has dimension 1.

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Let denote the vector space of polynomials of degree at most 3 in one variable and a collection of distinct real numbers. If the function defines and inner product on if

	a.
	b.
	c.
	d. for any positive integer .
	e. For no positive integer .

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If denotes the vector space of continuous functions on the interval and for we define the function For which positive integers does define an inner product on ?

	a.
	b. .
	c. .
	d. For all .
	e. For no positive integer .

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If denotes the vector space of continuous functions on the interval and are distinct points in , for which values of the positive integer m is the function





an inner product on ?

	a.
	b.
	c. all positive integers
	d. There is no such that is an inner product.

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If denotes the inner product space of continuous functions on the interval with inner product defined as ,


the orthogonal projection of the function onto the subspace spanned by equals

	a.
	b.
	c.
	d.
	e. None of the above

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True or False: Two non-zero vectors can span a 3-dimensional vector space.

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True or False: Any set of 4 vectors in a 3-dimensional vector space must be linearly dependent.

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True or False: Any set of 4 vectors in a 3-dimensional vector space must be linearly dependent

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