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 |

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

Choose one answer.

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 |

Similar matrices always have which of the following in common?

Choose one answer.

a. Column space | ||

b. Determinant | ||

c. Nullspace | ||

d. Eigenvectors | ||

e. Inverse |

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

Choose one answer.

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 |

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

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

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 |

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

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

Answer:

True
False

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

Choose one answer.

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

Answer:

True
False

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

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. |

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 |

Choose one answer.

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

,

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

Choose one answer.

a. The determinant of is . | ||

b. has rank | ||

c. The determinant of is . | ||

d. is symmetric | ||

e. is orthogonal |

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

Answer:

True
False

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

Answer:

True
False

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

Answer:

True
False