Notes on Linear Algebra

Warning for people trying to learn from these notes: I do linear algebra in a somewhat nonstandard way. Most linear algebra courses use the real and complex numbers. I use other number systems, which I describe in the first section on rings. You can still read a lot of the notes if you are just familiar with the real and complex numbers. If you notice that an example or discussion talks about "Z2" or "Z5" or something like that, it's a tipoff that it uses the "nonstandard" stuff. If you're taking a course which just uses the real and complex numbers, you could just ignore those examples or discussions. (Or you could read everything and surprise your instructor!)

A lot of the stuff that follows is directed toward instructors.

These are notes for a course in linear algebra. Students taking this course at Millersville University are assumed to have had, or be currently enrolled in, Calculus 3. Our Calculus 3 course covers vectors in 3 dimensions, including dot and cross products. Hence, my notes don't spend much time on basic vector arithmetic, since it would be redundant for our students. I may eventually say something about cross products as a digression in the sections on determinants, but right now they are not covered. Dot products occur as a special case of inner products in a later section. There are several sections on this material in my Calculus 3 notes.

There are several things about these linear algebra notes that are a little unusual. The most significant is that I've done as much linear algebra as possible over fields of nonzero characteristic. In practice, I confine myself to examples over the integers mod 2, 3, and 5, but I think this is enough to get the point across. To this end, the first section discusses commutative rings with identity and fields.

(This may worry some instructors, but after teaching this way for many years I found it isn't a big deal. Students don't seem to have that much trouble with modular arithmetic, given some practice. And it makes a lot of the matrix computations very clean and simple when you're working over (say) Z3! I also think modular arithmetic deserves to be more widely taught. A typical proof of something in linear algebra where the ground ring is a commutative ring with identity rather than a field uses algebraic facts like commutativity of addition that everyone takes for granted. So if you just do the proof without belaboring the fact that the ground ring isn't a field, it doesn't cause any confusion or puzzlement.)

Determinants are defined axiomatically, over a commutative ring with identity, and I think I've given complete proofs of all the results. The axiomatic presentation makes sense to me (probably because that was the way I learned linear algebra from the text by Hoffman and Kunze as an undergrad). It's a good opportunity to show students how an object can be defined by properties. Then you show it exists by construction, and finally you show it's unique.

Doing determinants over a commutative ring with identity is actually no harder than doing determinants over a field: The algebraic manipulations in the proofs rarely make any reference to the number system you're working in. Also, you will later use determinants with entries in a polynomial ring to compute eigenvalues. I've given a lot of computations of determinants (via row reduction and cofactor expansion) with all the steps shown.

In general, I feel having careful proofs and axiomatics in the notes is okay as long as there are lots of examples, and as long as you cover the rigorous stuff in class only as appropriate for the particular course. You are not required to say or read everything! I try to give enough examples with all the details shown that, even in the worse case if someone ignored all the proofs, they would at least be able to learn how to do computations.

Finally, the algorithm for the computation of the exponential of a matrix uses an approach which I believe is due to Richard Williamson. Of the approaches I've seen, it seems to be the simplest for hand computation, and (except for computing the eigenvalues, which is always an issue), can be implemented in computer algebra systems with little difficulty. If you haven't seen it, take a look - it's really neat!

[May 10, 2023] The notes have been revised through the section on null space. In addition, I revised the notes on the matrix exponential to fix some typos and improve the wording.

When the notes are in better shape, I would like to add exercises. (I have some from the courses I taught, but I need to revise them and add more.)

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