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.

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