M 341Syllabus

LINEAR ALGEBRA AND MATRIX THEORY

Prerequisite and degree relevance:
M408D or the equivalent or consent of instructor. (Credit may not be received for both M341 and M340L.
M341 is required for math majors; furthermore, math majors must make a grade of at least C in M341.)

The department allows instructors to choose between two texts. Whichever text you choose, it is recommended
that you read both syllabi, as each represents a different view of the course. The choice of text constrains
which topics you can reasonably cover, affecting in particular determinants and eigenvalues.

Primary Text - Andrilli & Hecker, ^{2nd edition}

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.

Suggested Coverage:
Chapter 1 - 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.

Chapter 2 - Cover all sections but again move reasonably to have enough time for Chapters 4 and 5.

Chapter 3 - For those instructors familiar with the first edition, it should be noted that this chapter has been
redone and the unusual treatment of determinants has been replaced by a more conventional one.

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 new and is a fairly reasonable attempt to introduce eigenvalues before introducing linear
transformations. The instructor should cover at least part of this section, all if desired.

Chapter 4 - This chapter is the meat of the course and the instructor should plan to take a good deal of time
here. Sections 4.1-4.4 should be covered thoroughly. The material in Section 4.5 is also important; however, it
is probably the most poorly written section in the book. An alternate presentation is recommended. Sections
4.6 and 4.7 should also be covered.

Chapter 5 - In a perfect world, the entire chapter should be taught. Realistically, at least Sections 5.1 and 5.2
should be covered. Heitmann 8/4/990
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