LINEAR ALGEBRA AND MATRIX THEORY
Prerequisite and degree relevance: M408D or the equivalent, with a grade of at leasr C, or consent of instructor. (Credit may not be received for both M341 and M340L. Majors with a 'math' advising code must register for this rather than for M340L; majors without a 'math' advising code must register for M340L. Math majors must make a grade of at least C in M341.)
Primary Text - Andrilli & Hecker, Elementary Linear Algebra third edition
Responsible Party: Ray Heitmann, January 2008
This course has three purposes and the instructor should give proper weight to all three. The students should learn some linear algebra - for most of them, this will be the only college linear algebra course they take. This is one of the first proof courses these students will take and they need to develop some proof skills. Finally, this is, for almost all students, the introductory course in mathematical abstraction and provides a necessary prerequisite for a number of our upper division courses. To teach this course successfully, the instructor should establish modest goals on all three fronts. On one hand, a student should not be able to pass this course simply by doing calculational problems well, but on the other hand, overly ambitious proof and abstraction goals simply discourage teacher and student alike.
To teach proofs, the instructor should cover Section 1.3 thoroughly to introduce various proof techniques. Afterwards, a liberal (but not overwhelming) number of proofs should be sprinkled in the lectures, homework, and tests.
In teaching abstraction, it is critical to remember that almost no students are capable of becoming truly comfortable with it in a single semester; it is self-defeating to establish this as a goal. The study of abstract vector spaces is a unified treatment of various familiar vector spaces and students in this course should never be taken very far from the concrete. Linear algebra is the perfect subject for teaching students that abstraction can be a friend. For example, it underlines nicely how the solutions to a homogeneous system are better behaved than the solutions to a non-homogeneous system. However, amusing examples of unnatural algebraic systems that may or may not be vector spaces should be avoided.
A warning should be given concerning the calculational homework problems. The authors, intending the students to take full advantage of technology, have made no effort to make problems come out neatly.
Chapter 1 Nine or ten lectures.
The first two sections provide necessary definitions for Section 1.3. The entire chapter should be covered. Generally move quickly but cover 1.3 meticulously. Three or four lectures should be devoted to this section.
Chapter 2 Six or seven lectures.
Cover all sections but again move reasonably to have enough time for Chapters 4 and 5.
Chapter 3 Three lectures.
Row operations are easy for them and you can go quite quickly here. Cover Sections 3.1 and 3.2. Section 3.3 is optional - you might also choose to cover parts of this section. Section 3.4 is a fairly reasonable attempt to introduce eigenvalues before introducing linear transformations. It is the interesting and important part of this chapter, at least in my opinion. The instructor should cover at least part of this section, all if desired.
Chapter 4 Fourteen or fifteen lectures.
This chapter is the meat of the course and the instructor should plan to take a good deal of time here. Sections 4.1-4.6 should be covered thoroughly. Section 4.7 is optional and should probably be skipped to provide more time for Chapter 5.
Chapter 5 About five lectures.
In a perfect world, the entire chapter should be taught, but 5.5 is probably too much to hope for. Realistically, at least Sections 5.1 and 5.2 should be covered.