←back to thread

The Little Book of Linear Algebra

(github.com)
468 points scapbi | 6 comments | 02 Sep 25 14:17 UTC | HN request time: 0.001s | source | bottom
Show context
andrewla ◴[02 Sep 25 18:02 UTC] No.45106778[source]▶
It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math.

But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matrices and Gaussian elimination, all the while bored out of your skull, until you have the tools to really start to approach the interesting areas.

Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks.

This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains.

It's funny because it's just a pedagogical nightmare to get students to care about any of this until one day two years later it all just makes sense.

replies(20): >>45106971 #>>45106998 #>>45107045 #>>45107321 #>>45107368 #>>45107522 #>>45107638 #>>45107686 #>>45108661 #>>45108864 #>>45109019 #>>45109881 #>>45111057 #>>45112025 #>>45112656 #>>45112834 #>>45113034 #>>45113260 #>>45114881 #>>45126195 #
1. Chinjut ◴[02 Sep 25 18:17 UTC] No.45106971[source]▶
Why do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning.

It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".

replies(2): >>45107074 #>>45107624 #
2. andrewla ◴[02 Sep 25 18:26 UTC] No.45107074[source]▶
Sure, that's a way to approach it. All you have to do is stay interested in "linear functions" long enough to get there. It's totally possible -- I got there, and so did many many many other people (arguably everyone who has applied mathematics to almost any problem has).

But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by". I stuck to it because that was what the course taught, and they wouldn't teach me multidimensional calculus without making me learn this stuff first, but it was months and years later when I suddenly found that linear functions were everywhere and I somehow magically had the tools and the knowledge to do stuff with them.

replies(3): >>45109862 #>>45110909 #>>45111274 #
3. cassepipe ◴[02 Sep 25 19:07 UTC] No.45107624[source]▶
Agree.The fact that it's just linear functions is what made it click for me
4. JadeNB ◴[02 Sep 25 22:20 UTC] No.45109862[source]▶
> But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by".

This seems to make it good motivation for an intellectually curious student—"linear functions are the simplest, dumbest kind of function, and yet they still teach us this new and exotic kind of multiplication." That's not how I learned it (I was the kind of obedient student who was interested in a mathematical definition because I was told that I should be), but I can't imagine that I wouldn't have been intrigued by such a presentation!

5. ndriscoll ◴[03 Sep 25 00:33 UTC] No.45110909[source]▶
Linear functions are the ones that we can actually wrap our heads around (maybe), and the big trick we have to understand nonlinear problems is to use calculus to be able to understand them in terms of linear ones again. Problems that can't be made linear tend to be exceptionally difficult, so basically any topic you want to learn is going to be calculus+linear algebra because everything else is too hard.

The real payoff though is after you do a deep dive and convince yourself there's plenty of theory and all of these interesting examples and then you learn about SVD or spectral theorems and that when you look at things correctly, you see they act independently in each dimension by... just multiplication by a single number. Unclear whether to be overwhelmed or underwhelmed by the revelation. Or perhaps a superposition.

6. bananaflag ◴[03 Sep 25 01:28 UTC] No.45111274[source]▶
Yeah, concepts can make a student reject them with passion.

I remember in a differential geometry course, when we reached "curves on surfaces", I thought "what stupidity! what are the odds a curve lies exactly on a surface?"