Notes on Linear Algebra

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.

There are several things about these linear algebra notes that are a little unusual. One novelty 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 isn't a big deal. 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 in, from 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. But it's also a matter of intellectual honesty, since you will later use determinants with entries in a polynomial ring to compute eigenvalues.

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 (modulo computing the eigenvalues, which is always an issue), can be implemented in computer algebra systems with little difficulty.

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