The latest book I’ve been reading is “Algebras of Linear Transformations”, by Douglas Farenick. The first two chapters are supposed to be a review of linear algebra, but I’ve already picked up a lot of interesting ideas from the first quarter of chapter one. Not necessarily new ideas, but ones I haven’t seen in a while, and certainly not in this interesting a light.

Here’s an example: to prove the rank-nullity theorem, the author takes a much more scenic and abstract route than Hoffman and Kunze— the authors of the infamous “Linear Algebra” textbook we’re using in my advanced linear algebra class. First he shows that if is a subspace of , then , which is pretty intuitive, then he shows the first isomorphism theorem: if is a linear operator, then , which is again intuitive. From there it’s a short step to take to get the rank-nullity theorem; for me this exposition has the advantage of introducing two results that are interesting in their own right, and then showing how they are not only interesting, but also useful. Wonderful stuff.

One more example:

Let be a linear functional on .

1. There is a unique matrix such that for all
2. If for all , then there is an such that for all

This is a great result, because it gives one use for the trace operator— determining functionals. It’s also interesting in that it provides another way of determining functionals, as opposed to inner products via the Reisz Representation Theorem.

Next are inner product spaces, which I look forward to with glee…

Possibly relevant posts:

• Product sigma algebras (9/26/2008)
• Distribution function (1/14/2006)
• Stochastic Lyapunov functions (5/6/2007)