Due: Right After Spring Break
Pettofrezzo text: 1.1 - 2, 4, 6,
7.
1.2 - 2, 3, 4, 7, 8, 10, 12. 1.3 - 2, 4.
Rosen
text: 3.8 - 2a, 4a,c, 5, 10,
18, 20, 21, 23, 24.
My problem:
You are asked to find out how many rectangles there are in an nxn grid of rectangles, and how many squares. These two problems can be solved with methods from discrete math I, and you did them for homework last semester. Now let's solve them with linear algebra, assuming the closed form for the answers are polynomials (this assumption happens to work, but we would normally have to justify it somehow).
For each problem:
a. Using data you gather by hand, find out
the answers for n = 1 to 5.
b. Using finite differences, calculate what
degree polynomial r the answer should be.
c. Using the polynomial Arxr+
Ar-1xr-1+ ... + A0, and r+1
data points, constuct a set of r+1 equations.
d. Solve the system of equations, to calculate
the two closed form polynomial answers. Check them against your
direct
solutions from last semester.
The point of this problem is to show that 2
points
determine a line, 3 points determine a parabola, and r points
determine
an r-1 degree polynomial. The exact calculations use
basic
linear algebra to solve a system of r equations.