I received an email yesterday afternoon from an incoming freshman at the University of Toronto. You could tell that they were very excited to start and they couldn't wait to get their hands dirty with mathematics. They would be taking the advanced linear algebra course designed for the math specialist program and they wanted my opinion on how to prepare for it. My first reaction was to ignore the email. But then I remembered stories of Ragnar-Olaf Buchweitz told by Graham Leuschke and Michael Wemyss last month and decided to write to them. It turned into a blog post.

At the University of Toronto, I taught a lot of linear algebra courses. I TA'd MAT223 - your standard linear algebra/matrix theory course three times. The continuation of MAT223, that is MAT224 also three times. Then, I TA'd MAT240 - the advanced linear algebra course and its continuation MAT247 three times. On the other hand, I taught MAT223 once and I coordinated MAT224 twice. That is a lot of linear algebra.

The math major courses MAT223/224 are standard linear algebra courses. The material is the same in almost every university in the world. You could pick a test from an equivalent course in an arbitrary university in Canada and most of the questions will be suitable for exam questions in these courses. On the other hand, MAT240/247 are quite challenging courses, to be honest. But they also produce very good mathematicians.

I must say that I enjoyed TA'ing MAT240/247 most. Surrounded by very enthusiastic people, I was able to teach a lot of cool stuff in tutorials. Within the first month of their university life, not too many people see quiver representations or cohomology computations. I admit that I spent only a small portion of my tutorials on these and only a small portion of students actually understood what's going on, but somehow almost everyone was super excited about it every time I did this.

So, what can you do to prepare for MAT240/247 and to be successful in these courses?

Linear algebra is a very useful tool. Methods and concepts you learn in this course are going to be very useful in the future. However, what I value most about this course is that it is an introduction to how to be a mathematician. A linear algebra course is very suited for this because we understand linear algebra very well and the results are usually the best you could hope for. The proofs often follow a nice structure. The story is complete. Perfect landscape to teach how to do math.

Hence, the last thing I would recommend for preparing for this course is to look at a linear algebra book. First focus on fundamental skills.