alt Hacker NewsEvery time this subject comes up, or really any math subject comes up, someone recommends 3Blue1Brown. I love 3Blue1Brown. Just like when The Shawshank Redemption plays on TBS for the 389248th time, I will stop what I'm doing and rewatch any 3Blue1Brown video as soon as it appears in my feed.
But I'm not sure I've ever really learned anything from one of those videos. Appreciated something more? Absolutely. And maybe, sure, that's a kind of learning. But I cringe every time eigenvalues come up and people point to the 3B1B evector video.
In fact, if your goal is to actually get any kind of facility with the concept, this "weaksauce" blog post probably has a better didactic strategy than 3B1B. It strips the concept down, provides specific, minimized worked examples, and provides a useful framing for the concept (something basically at the core of 3B1B's process).
I learned linear algebra from Strang's 18.06. I later did a bunch of Axler helping my daughter through UIUC linear algebra. I like both. Strang is much closer to what the median HN person probably wants. In both cases though: don't do what I did at first, and just watch the videos and read the book. If you're not doing problems, you're probably not learning anything.
This blog post comes closer to "actually doing problems" than 3B1B. Ergo: its sauce is stronger, not weaker.
I came to the blog post expecting to roll my eyes. No discussion of inner product spaces? Not even a mention of conjugate symmetry? I was pleasantly surprised.
It's not easy to come up with a simple, accessible framing for a topic like this, and, maybe, the dot product is particularly tricky to give an intuition for (I'll go out on a limb and say that neither Strang nor Axler do a particularly great job at it --- "it" being, explaining the "why" of the dot product to someone who doesn't really even know what a vector is). The post doesn't purport to teach all of linear algebra. It's just an exercise in trying to explain one small part of it.
I'm not asking you to give the author a break, so much as suggesting that you're closing yourself off from appreciating different strategies for explaining complex topics, which is a valuable skill to have.
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When I was in school, my teachers and professors all insisted that the only way to learn a subject was to _practice_ it. If you want to learn math, work the problems. I was skeptical when I was young, but I have yet to find any other way to learn math (or anything else).
But at the very least, surely a dot product explainer should talk about the two main ways of looking at a dot product! This article leaves out the "angle and norms" (||a||·||b||·cos(θ)) interpretation entirely. It's like if someone gave you the formula for complex multiplication, without also showing you how it's all about rotations.
And maybe I'm being a pedant, but the dot product should be between two things that are in the same space. In the Minnesota lottery example, the probabilities should be a row vector instead. It's the exact same calculation, so again, maybe a bit too pedantic.