## 18.06 Spring 2006 : Problem Set #1

### due 4PM Wednesday 2/15

(Asterisk means the solution is in the back of the book)

Section 2.2: 19, 21, 26*

Section 2.3: 3*, 17, 27*

Section 2.4: 2*, 24, 33*, 34*

Section 2.5: 7*, 9, 28, 29, 30

MATLAB (or some other computer language) Problem: This variation of
problem 30 in Section 2.2 asks you to take A = randn(3,3). Note the
``n'' in randn for the normal distribution. Compute the square of the
product of the three pivots. Find experimentally this average
square. Nice answer, isn't it? NOTE: A(2,:) is the second row of A,
etc.

Hint: To see the nice answer, do not use any pivoting strategy that
does row exchanges. MATLAB has a high quality solver called LU which
does incorporate this strategy.
Better for education to perform (row = row - multiplier*row) yourself,
but in fact prod(diag(U))^2 does give the right answer from LU.

Extensions (not for credit): Experimentally try out n = 4 or n = 2 and
guess the general formula.

RESEARCH PROBLEM (no points, but if you have the right skills, and
want to impress and just for fun): Somewhere on the web, maybe there
are the Red Sox/Yankees games box scores for the last, oh, 5
years. Assuming that every game may be thought of as a pair of 9
element vectors (dropping extra innings, putting in 0 if appropriate),
count how many dot products are 0, 1, 2, .... Don't ask your TA
whether it's okay to do 7 years, or to use football, or some other
real world thing (sure why not) just if you feel like having fun, let
us know what you find by writing it down.

* Copyright © 2003 Massachusetts Institute of
Technology*