Math 2573H Honors Calculus III Fall Semester 2002
Assignment 2 - Due Thursday 9/19/2002
Read: Williamson and Trotter, Chapter 3, Sections 4 and 5.
Section 5 is quite substantial and I hope we can get through all this in a week.
Chap 3, Sec 4 (pages 78-80): 1a, 1b, 2, 3, 3b*, 4*, 5*, 8, 13.
Chap 3, Sec 5 (pages 88-89): 1b, 2*, 5, 6*, 8a*, 10*, 11a, 11d, 11e.
You have quite rightly been asking what the point of matrices is, and why we need to go to a lot of trouble to introduce definitions and jargon, when it seems we can perfectly well do our calculations without knowing about matrices. After all, as far as solving systems of linear equations is concerned, you can all do this by your own perfectly good methods, and you could already do it before you started attending this semester's class.
Your comments have made me also think about why we introduce matrices. I suppose the fact is that matrices are primarily a notational convenience, at least at the beginning: instead of writing down a system of equations we only write down the coefficients. However, in sections 3, 4 and 5 of Chapter 3 we start to see more and more complicated operations which may be performed on matrices. We see matrices being regarded as algebraic objects in their own right, which can be manipulated in the same way as ordinary numbers. This is a conceptual step in which we start to denote matrices by single symbols such as A, B, C etc. The algebra of matrices is introduced because it is convenient and useful, in the same way that we find it convenient and useful to do arithmetic with ordinary numbers, and it would be hard to study the algebraic structure of matrices without having the notion of a matrix in the first place. It is too much to expect to see the full implications of this structure all at once, but we can see at least some of the applications. One of them is the approach to solving a system of equations using the inverse matrix, described on page 74, which would be hard to think of without the notion of a matrix.