Linear algebra self-study

I finished self-studying Axler’s Linear Algebra Done Right (3rd edition). I wanted to understand linear algebra and picked up LADR because it’s the most recommended book online. Some notes below.

After picking the book I found a corresponding syllabus online. I wanted one that has a clear reading schedule and homework assignments. The one I initially used was this one from Brown. I tried to follow the actual class schedule, but dropped that pretty quickly. Sometimes I ended up going faster, sometimes slower. I finished the course in three months– on balance about the same time as the original class schedule. The Brown syllabus didn’t cover the entire book, but I wanted to do more chapters. So when I was done I found another syllabus from Berkeley that covers more of the book, and did the remaining problems from there.

I did all the homework problems in Overleaf. There were a total of 197 problems assigned; I ended up finishing 167. There were 30 problems I couldn’t solve quickly enough and decided to move on. I’d like to come back to these, but have not done that yet.

I extensively used ChatGPT 4 to check my homework problems (the paid version; I found ChatGPT 3.5 isn’t good enough). Prompts like check the following homework problem” or critique the following proof” usually produce very good results. Sometimes ChatGPT was wrong in its critique. Typically, but not always, tinkering with the arguments to make ChatGPT happy made for better proofs anyway.

One edge case where ChatGPT performed poorly was proofs by contradiction. It tends to have a lot of trouble understanding those. A minor nit is that ChatGPT UI renders Latex correctly maybe half the time. When it does, the experience is excellent. When it doesn’t, reading the output is a pain.

Sometimes I needed hints to solve homework problems. For those I texted a friend with a math degree and he’d usually point me in the right direction (thanks Ryan!) I never found a prompt that would get ChatGPT to give good hints; my friend’s hints were always dramatically better.

In general I find myself bored with lectures, I almost always would rather read a textbook. But there were a few points where the material got especially difficult. For those I supplemented with these lectures from Penn State that are pretty faithful to the book. I must admit to occasionally phoning it in on understanding some proofs. Axler’s proofs are beautiful, but his proofwriting was a hit and miss for me. Some proofs flowed like a poem. Others sputtered. It was never too hard to understand the sputtering ones, but it did require a degree of conscientiousness I sometimes wasn’t able to generate.

Which brings me to the book itself. I have mixed feelings about LADR. On the one hand, Axler delivers. If you diligently read the book and struggle through the exercises, you will understand the material. And once you’ve understood the material (and often even before that) you can appreciate the elegance of the exposition.

This commitment to elegance makes the material more difficult to absorb. Some struggle is endemic in learning math, but it need not be any more difficult than necessary. Later I found Terence Tao’s linear algebra notes, which are roughly as rigorous as LADR but are much easier to understand. One plausible reason is that I found Tao’s notes after I’d already worked through LADR and understood the abstractions. But I don’t think so. It seems to me that Tao set out to minimize student confusion and Axler set out to write an elegant linear algebra book. Both texts achieve their respective goal.

May 12, 2024