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Comment by id_ris | Hacker News Reader

I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head.

Now that the ideas of things like vector spaces, norms, orthogonality, rank, basis, etc are nearly second nature, the concepts are useful as I study other branches of math which would feel impenetrable otherwise.

YMMV, and if you can learn from condensed materials go for it, but I might be too dumb for it work lol. I think the real benefit accrues to the author who had to work out how to teach these concepts to others.

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My experience from a decade of doing professional maths is that there are no shortcuts. You only learn maths by doing hundreds (thousands) of exercises, both mundane and more exctiting.

Also, the concepts "mature" in the brain. I remember sleepless nights in the first year of undergrad spent on understanding the details of the proof of the Jordan decomposition and a few years later (when studying algebraic groups) it all felt trivial.

There's no shortcut to understanding maths, just a lot of time spent in solitude trying to make sense of all these abstract concepts (and they DO make sense).

I think there is a shortcut: use examples (a "domain") rather than starting with theory. learning something purely in the abstract is easy to forget.

You don't learn by putting information into your head, you learn by retrieving information from your head.

I agree, and have to admit that my own knowledge of LA is sadly way too superficial. The article did give a me a big lightbulb on something I didn't understand before: Some of my friends work in quantitative sociology and economics, and use Stata for their programming needs. They basically use matrices as their exclusive data structure, and the reason for that eluded me until after reading this article.

> While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head.

I'm not sure I understand your point. Are you just saying that this blog post isn't an adequate substitute for taking a course in linear algebra? (Of course it isn't. But who said it was?)

Unfortunately, for a lot of people, including undergraduates, the dopamine hit they get from watching a video or passively reading a textbook makes them believe that these are adequate substitutes for doing thousands of exercises.

In my university, undergraduates have admitted that they have done fewer than 50 questions throughout the entirety of my math course. Their grades obviously reflect that, but they will do the same next semester.

Can I press you to explain what you think dopamine is/does?

I'm curious because I'm a neuroscientist who occasionally works on dopaminergic stuff and "passively reading a textbook" is so far from the canonical examples we use for dopamine activity, but the idea of dopamine/dopamine 'hits' has taken on a life of its own that seems quite different from the neurotransmitter.

I am a physicist, so I will dare not make any technical neuroscience claims. I meant "dopamine" in the causal sense of people feeling pleasure from passively reading a book because they think it is useful work.

There is a lot of excitement around machine learning and AI and there is not a proportional excitement for the math that underlies the theory.

In a lot of content that teaches machine learning/AI, the linear algebra substrate of it is given short shrift.

The consumers of that content infer that the backing mathematics is easy or unimportant, while the creators of the content are not actually implying that, but just want to move on to what the audience came for.

I'm not an expert in machine learning so I can't say whether you can get by without a strong understanding of linear algebra, but my intuition answer is no you can't. Beside that, I enjoy the math for its own sake so I'm happy to trudge through the textbooks.

And no, I don't think the blog post makes any claims that it is a subsitute for taking a course in linear algebra.

3blue1brown himself has told in his videos that they are not substitute for books and working out the exercise and he recommends reading from books and his videos are for inspiration and as a supplement.

> I think the real benefit accrues to the author who had to work out how to teach these concepts to others.

In college I learned more math when I was trying to build software for teaching math compared to when I was trying to learn math.

Yeah, this sort of stuff is fine supplementary material... but nobody really learns anything from a blog post or some videos.

I disagree. Maybe you'll remember it not as well if you don't do exercises, but there is no reason you can't learn from a blogpost (which is trivial to prove since you can just copy-paste the contents of a book in the form of one or more blogposts).

What your parent comment meant was that it is not possible to learn well from typical blog posts like this that try to condense the subject into a 3000 word article. Of course, if you copy-paste the content of a book into a blog post, then parent comment's point no longer applies.

I'd say it still applies, even when reading a textbook going through the exercises is crucial

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