Math 104.01 (Linear Algebra)

Spring 1999

Plan for Week 6

We start this week with our last look at Chapter 1, in which we practice modeling, that is, converting real-world contexts into systems of linear equations. The contexts developed here will be important for, among other things, Labs 7, 8, and 9.

The main topic this week is DETERMINANTS.This subject is a little different from everything else we will do this semester. Your author notes that determinants used to be much more important before we had computing equipment and large matrix applications. The subject has lost much of its practical application, but it's still important from a theoretical perspective. Our classroom work, homework, and lab will all be related, and computer experiments in the lab will help nail down all the important properties of determinants.

To see the syllabus for Week 6 in a separate window, click here.

Notes:
  1. Your next homework papers will be turned in on Monday, Feb. 22. Those papers should include solutions to all problems in the assignment below. The assignment dates are start dates.
  2. In general, no solution will be given full credit unless you have written an explanation of why you know it is correct. (Exceptions to this rule are the exercises whose numbers appear in parentheses.)
  3. Section 1.8 is not included in our syllabus. For present purposes, the only matrix A that will be assigned to a linear transformation T is what the text calls the "standard" matrix, that is, the one such that T(x) = Ax for all x in the domain of T.
  4. In section 1.9 your author is a little careless with significant digits. A case in point is Example 3. We don't know how many digits are significant in the original population estimates -- it looks like only one in each number, as in 6x105 people in the city. To be charitable, let's suppose that the initial populations are known to the nearest thousand, i.e., to three significant digits (3SD). Then the figures reported for 1991 make sense, but the ones for 1992 do not. The calculations are all right, but the 3SD answers should be reported as [565,000, 435,000]. The point is that doing calculations on data cannot add significance that was not present in the data. Be careful about this -- even when your author is not.
  5. Continuing the point about SD, suppose the original populations in Example 3 are known only to 1SD. The calculations are still all right -- there is no need to discard digits in intermediate calculations -- but now the answers should be [600,000 400,000] in both years. That is, the changes are slow enough that they are invisible at the 1SD level in such a short time frame. To see the short-term effects of these migrations, one would need better data.
  6. In Exercise 11 on p. 94, the phrase "significant decimal places" is used. This means significant digits, not places after the decimal point.
  7. None of the computer exercises in the text are assigned this week because they all overlap the lab. Similarly, Section 3.3 is assigned as reading only because the important part of it is all contained in the lab.
  8. Your instructor will be away from Wednesday morning through Sunday, February 21. Class on Wednesday and lab on Friday will be covered by a substitute.

Assignments

David A. Smith <das@math.duke.edu>

Last modified: January 23, 1999