# A Small Personal Breakthrough: the real goal of linear algebra

Originally Posted 09-30-2012

Think of a course you are teaching right now. What is it really about?

I have been grading linear algebra homework tonight, and I finally had a substantive thought about what the class really should be.

Maybe I am funny, but I have a lot of trouble figuring out the point of the courses I teach, and until I do, everything is just a hash. Without a guiding principle, I have no decent way of making choices and structuring course work.

And I can’t seem to think up a reasonable answer to the question “What is this course about?” until sometime in the middle of the second or third iteration (at the earliest). Apparently I am slow. But it only really dawned on me that I needed this information when I figured it out for Euclidean Geometry a few years ago.

For example, my Euclidean Geometry class is really about the nature and role of mathematical definitions, and how that gives us a foundation for the axiomatic method. It took a few semesters to realize that this is what the students didn’t know and what I was really helping them learn.

I think that linear algebra is about the process of making your own examples and abstracting out the patterns. Seriously. Undergraduate linear algebra is all about this “simple” process of solving systems of linear equations using Gaussian Elimination. The rest of the semester is about context, motivation, interpretation and abstraction of that basic process. And it seems the the mathematical skill that my students lack is the ability to make their own examples for intellectual profit.

Well, it is not that they don’t know how to make an example. It is that they don’t seem to understand that this is a foundational skill. What do you do when you have a new term to comprehend? Or a theorem to understand? Make Examples.

Linear algebra seems like a good place to teach this. The whole course can be understood by having enough examples at your fingertips and learning to use them as prototypes. All of the crazy abstraction is just organizational language for how to apply the prototype examples.

Tell me, am I crazy? What do you all think? If this works, I have a way to organize and refine my course. The next time through, everything will go much smoother because I’ll know how to orient myself. I can make choices with the main goal in mind.