Returning to linear algebra
Mar 14, 2024
My undergraduate degree was in pure mathematics. After a rocky start, I really grew to love it, by third year regularly writing a math blog that documented the cool things I was learning. I definitely was pretentious about it, and it would be untruthful to say that some part of my enjoyment didn't derive directly from feeling intellectually self-important, but most of my enjoyment really was inherent.
In my math program (and indeed in most undergraduate math programs), the first proof-based course students take is linear algebra. This course was a struggle. I didn't know how to write or structure proofs, but more fundamentally I didn't grasp the point of proofs. Many times, we were asked to prove statements that looked pointless to me, for example, to prove that for a linear map T: V → W, T(0V) = 0W. Isn't it obvious? A linear map of zero returns zero: mx + b is 0 if you plug in zero??? In any case, without getting too in the weeds, what I'm trying to say is that I deeply learned very little, if anything, during this course. Ever since, I've had the uncomfortable feeling that I'm missing something fundamental to my understanding.
Of course, linear algebra is very important in many mathematically-based areas of study: engineering, statistics, and physics are just a few. I recently encountered something relying on linear algebra in my reading, which reminded me of my undergraduate class in the subject. Curious on how I might read it differently now, I tracked down the textbook we used and read a few chapters. I was pleasantly surprised! Many of the chapters that I found mystifying were perfectly clear now.
As a concrete example, the purpose of the proof exercise I mention above is now immediately apparent to me, because I have background context from an abstract algebra course I took later on in the degree. It's just asking for you to prove that the image of 0V satisfies the properties of the zero vector in the vector space W. Moreover, I understand how to write the proof, which statements I'm allowed to use to make the argument, and in which order to put them to get a proof. I know the "rules of the game", so to speak.
It's funny how our brains can sometimes much better grasp things once we have more context and practice. In particular, I think three things really helped: first, and most important, I now have familiarity with proofs and a better conceptual understanding of them; second, I took (and very much enjoyed) abstract algebra later in my program, which is extremely helpful to frame linear algebra; and third, I have both much more mathematical confidence and also much more context on why we might study linear mappings, which comes with better intuition on how linear algebra operates. Even though I went through years of mild insecurity about my perceived deficiency of linear algebra, I was more solid than I thought. It's as though my mind was making backwards connections subconsciously, prospecting through what I didn't even know I remembered from that introduction to linear algebra.
Because of this experience, I think we may be giving our minds and ourselves less credit than they deserve. Maybe they have, in the background, worked through something that you didn't understand fully the first time you saw it. But I have also learned that it's an even better idea to go back and review whatever the topic is to complete the loop. You get to explicitly solidy your understanding of it with those connections, and you also get to update your self-perception of how well you know it.
There is still one key chapter I haven't revisited yet, which is on inner product spaces. Maybe I will get to it and write a blog post on that, too!