Math 215 Section 2 Second Hour Exam November 8, 1995

Your Name:

The number in parentheses before each question gives its point value. The total is 100 points. If you want any partial credit, you must show your work.

1. (20) Consider the vector space P_{3} of polynomials of degree 3 f = a_{3}x^{3} + a_{2}x^{2} + a_{1}x^{1} + a_{0} (x a variable, a_{0}, …, a_{3} real numbers). Determine whether each of the following subsets is a vector subspace of P_{3}:

a) The set of f with f(1) = 1.

b) The set of f with a_{1} = 0 and a_{3} = 0.

c) The set of f with a_{1} = 0 or a_{3} = 0.

d) The span of x^{2} − x + 1 and x^{3} − x^{2} + x.

e) The set of f with \frac{df}{dx} = 0.

2. (10) Suppose A is a 3 × 3 matrix. You perform the following row operations on A:

(i) Multiply row 1 by 1/3 (ii) Subtract 7 times row 1 from row 3 (iii) Switch rows 2 and 3 (iv) Multiply row 2 by 4

The final result is:

( \begin{matrix}1 & 0 & 2 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix} )

What is the determinant of A?

3. (15) Let V be a finite dimensional vector space and let v_{1}, …, v_{k} be vectors in V. Explain what is the most information the following statements give you about the dimension of V. (Your answers should look like “dim V is bigger than —, or less than —, or equal to —,” or “no information.”)

a) v_{1}, …, v_{k} are linearly independent and do not span V.

b) v_{1}, …, v_{k} span V.

c) v_{1}, …, v_{k} are linearly dependent.

4. (20) Consider the following two bases of R^{2}:

B = \{[ \begin{matrix}1 \\ 1 \end{matrix} ] , [ \begin{matrix}1 \\ −1 \end{matrix} ]\} B′ = \{[ \begin{matrix}0 \\ 1 \end{matrix} ] , [ \begin{matrix}1 \\ 1 \end{matrix} ]\}

a) Which vector v in R^{2} satisfies [v]_{B′} = [ \begin{matrix}1 \\ −2 \end{matrix} ]?

b) Find the coordinates of [ \begin{matrix}2 \\ 3 \end{matrix} ] with respect to B′.

c) Find the change of coordinates matrix P_{B→B′}.

d) Find the coordinates of [ \begin{matrix}2 \\ 3 \end{matrix} ] with respect to B and check your answer against parts b and c.

5. (10) Suppose A and B are two square matrices of the same size and c is a scalar. Are each of the following statements always true or sometimes false?

a) det(AB) = det(A) det(B)

b) det(A^{t}) = − det(A)

c) det(AB) = det(BA)

d) det(A + B) = det(A) + det(B)

e) det(cA) = c det(A)

6. (25) Let A be the 3 × 4 matrix below. Find a basis of the null space of A and calculate its dimension. Also, find a basis of the column space of A and calculate its dimension. What relation should these dimensions satisfy?

A = ( \begin{matrix}−2 & 4 & −2 & −4 \\ 2 & −6 & −3 & 1 \\ −3 & 8 & 2 & −3 \end{matrix} )