Math 104.01 (Linear Algebra)

Spring 1999

Plan for Week 3

This week we formalize and put to work the concept of vector. Much of what we have to say about vectors (at least in two and three dimensions) will be review of information from some calculus or physics course, or perhaps a high school course.  However, we will also see a powerful new idea: treating the columns of a matrix as vectors and the matrix itself as a list of vectors.

This new idea leads us to treat a system of linear equations as a single matrix equation, Ax = b, a variant of the vector equation concept. In particular, we interpret this equation as saying that the vector b is a linear combination of the columns of A. Thus, asking for solutions of the matrix equation is the same as asking whether, in fact, b is a linear combination of columns of A, and, if so, in what ways.

In lab this week we jump ahead to Chapter 2 and explore the properties algebraic operations involving matrices. The relevance of these operations may not be immediately apparent, but they will play major roles in succeeding weeks. In general, to accommodate the subject matter of some of our labs -- while keeping them spaced at one-week intervals -- we need to intersperse sections in Chapters 1 and 2.  We will cover all the important concepts, but not exactly in the order contemplated by your author.

To see the syllabus for Week 3 in a separate window, click here.


Notes:
  1. Your next homework papers will be turned in on Monday, Feb. 1. Those papers should include solutions to all problems in the assignment below. The assignment dates are start dates.
  2. Your textbook has answers in the back of the book for odd-numbered exercises. If you look there first, you will subvert the learning process. If you know your answers are correct, you will never need to look there. 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.) For example, an acceptable explanation for a solution of a linear system of equations is that you have substituted the proposed solutions into the original equations and found that they satisfy all of the equations simultaneously -- and you have to show your work. Two examples of unacceptable explanations:

    Some exercises ask explicitly for an explanation -- in this case, you are not being asked to do more than the exercise calls for.

  3. Submit your completed Maple worksheet for Lab 3 as an attachment to e-mail by the end of the day on Wednesday, February 3.
  4. Remember to submit your e-mail journal entry on Friday, January 29.

Assignments


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

Last modified: January 5, 1999