Math 308C, Sample Problems

The midterm will be closed book, closed notes. No calculators will be allowed.

1. It can be shown that the maximum number of regions into which n planes divide 3-space is given by the following table (don’t try to prove this).

number of planes number of regions
1  2
2  4
3  8
4  15

For each n, call the number of regions f(n). Find a polynomial p(n) of degree three, p(n) = a + bn + cn^{2} + dn^{3} such that p(n) = f(n), n = 1, 2, 3, 4.

2. Find all angles 0 ≤ α ≤ 2π, 0 ≤ β ≤ 2π such that

cos α + sin β = 1
3 cos α − sin β = 1

3. For the following three vectors, decide whether or not there are scalars a, b such that S = aT + bU. If there are such scalars find them, if not say why not.

S = [\begin{matrix}3 \\ 4 \end{matrix}] , T = [\begin{matrix}5 \\ 6 \end{matrix}] , U = [\begin{matrix}1 \\ −1 \end{matrix}]

4. Find the numerical quadrature rule Q(f) = A_{0}f(1) + A_{1}f(3/2) + A_{2}f(2) for integration on the interval [0, 2], that is exact on polynomials of degree at least two.

5. Compute the inverse of

[\begin{matrix}2 & 5 & −1 \\ 0 & 4 & 1 \\ 0 & 0 & 2 \end{matrix}]

6. Let {V_{1}, V_{2}} be linearly independent vectors in R^{2} and let a_{ij} = V_{i}^{T} V_{j}.

Let

A = [\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}]

(a) Prove that A is symmetric.

(b) Prove that A is nonsingular. Hint: If Ax = 0, look at x^{T} Ax.

7. For what values of t is the matrix

[\begin{matrix}1 & 1 & −1 \\ 0 & 1 & 3 \\ 1 & 1 & t \end{matrix}]  (1)

nonsingular?

8. I may also put homework problems on the exam, so make sure you know how to do the homework.