08.21.2016
Dear Student,
As I mentioned in my "Instructor's Welcome to Math 488" message, our normal means of communicating with one another will be by email or by phone. I also said that will send you a "Math 488 Week" message at the beginning of nth-week. In that message, I will summarize the content of the lectures and homework for that week and outline how the lectures and homework for that week fit into the rest of the course. I hope that you will find these weekly messages instructive and helpful as you progress through the course.
You will see that each lecture in my course begins with a short review of the terminology and skills of previous lectures and then go on to some related graduate-level mathematics topic. This combination of review and augmentation of engineering-related mathematics persists throughout the course. The course textbook by Kreyszig is also very helpful for this purpose because it provides additional background information on most of the course topics including numerous examples and illustrative problems with solutions to guide you through the mathematical concepts and methods covered in the course.
Our class schedule specifies that you should complete Lectures 1, 2 and 3 and the associated homework by Sunday, August 28.
Week 1 -- Content Summary for Lectures 1, 2, and 3
Much of the content of these three lectures should be a review of things that you studied as an undergraduate student, such as solutions of small systems of linear equations, matrices, matrix operations, inverses and determinants. Consequently, we cover these topics rather quickly.
In Lectures 1 and 2 we discuss the important idea of an echelon form of a rectangular matrix and the special technique of row reduction to echelon form. These two concepts are basic to our discussion of linear independence of vectors, spanning sets of vectors and the general solution of homogeneous systems of linear algebraic equations. This content is important for its own sake but also because it sets up our approach to the solution of systems of linear differential equations in subsequent lectures.
We also set up the content for future lectures by discussing the fundamentals of vector spaces and linear operators on vector spaces. As a result, there is quite a bit of terminology discussed in Lecture 3 (and also Lecture 4 in Week 2). There is no need to memorize definitions in the abstract context of vector spaces. It is far more important to understand the meaning of the concepts in the context of ordinary vectors in a plane or space or matrices through concrete examples and problems. The examples and 'Just Do It' problems in the lectures, along with the problems in the homework, will help you to do this.
Be sure to listen to the Audio Clips and do the Just Do It problems in the lectures. They are used to introduce, summarize and explore the ideas in the lectures as you go along.
You should be able to do most of the homework problems, using only the Hint and checking it with the Answer, without relying on the Complete Solutions provided in the Homework facility. It will often be the case that a homework problem in Lecture n will be easier after you read the first part of Lecture n+1 because the first part of Lecture n+1 reviews the content of Lecture n and that helps you to complete the solution of a problem that you had difficulty with in Lecture n.
If you have any questions as you work through this week's lectures and homework, please contact me.
Best wishes for a good start with the course this week!
Tony Peressini
Professor of Mathematics (Emeritus)
Cell PH: 217 840 2871
anthonyperessini@gmail.com