# 28

3 blue 1 brown is a youtube channel that teaches math concepts. I've found it a much better introduction than other resources I've looked at. The Essence of Calculus series was particularly good.

But one issue is that, while the videos come with a few exercises sprinkled within (typically one per concept), they don't come with enough to really check whether I understand a thing.

Last year I tried binging the channel but kept running into issues where I'd want to do additional practice, so I'd try Khan Academy or Brilliant.org. But it felt like the exercises were introducing concepts in a fairly different order, or without using the same metaphors/explanations/keywords that 3-blue-1-brown was. So it was hard to tell what exercises corresponded with what. (Brilliant.org was better overall than Khan Academy but still felt like a completely different lesson plan, with somewhat less-good-explanations)

Eventually, bouncing between brilliant.org and 3 blue 1 brown resulted in me running out of steam and giving up.

I've emailed the creator about it but even if they were excited about it I imagine it'd be quite a while before anything useful came of it.

I'd be generally interested in a project of crowdsourcing exercises that correspond to individual 3-blue-1-brown videos. Since I'm currently trying to re-ignite my interest in the Linear Algebra series I thought I'd ask specifically about that, although if people happen to have cached thoughts about other series that'd be cool.

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If you want a proof-based approach, Linear Algebra Done Right is the typical go-to that's also on the MIRI page. I went through maybe the first 3/4ths of it, and I thought it was pretty good, in terms of number of exercises and helping you think about manipulating vector spaces, etc. in a more abstract sense.

Otherwise, I've heard good things about Gilbert Strang's MIT OCW course here: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/.

In general, I think that 3B1B's videos are really good for building intuition about a concept, but trying to do exercises off of the pedagogy in his videos alone can be quite challenging, especially as he often assumes some mastery with the subject already. (EX: In the eigen-stuffs video, he doesn't actually explain how to find the eigenvalues of a matrix.)

Thus, I think it makes more sense to stick to a traditional textbook / course for learning linear algebra and using 3B1B as supplementary stuff for when you want a visual / different way of looking at a concept.

Also, it might be worth checking in to see what you want to learn linear algebra for. I suspect there are more domain specific resources if, for example, you cared about just the useful parts of linear algebra used in machine learning (dimensionality reduction, etc.).

Previous LessWrong reviews of Linear Algebra Done Right by Nate Soares and TurnTrout (both highly detailed).

Here's a summary passage from Nate:

This book did a far better job of introducing the main concepts of linear algebra to me than did my Discrete Mathematics course. I came away with a vastly improved intuition for why the standard tools of linear algebra actually work.
I can personally attest that Linear Algebra Done Right is a great way to un-memorize passwords and build up that intuition. If you know how to compute a determinant but you have no idea what it means, then I recommend giving this book a shot.
I imagine that Linear Algebra Done Right would also be a good introduction for someone who hasn't done any linear algebra at all.

TurnTrout's review has links to solutions to the exercises (he did ~100% of the exercises, and talks about them a bit, which is relevant for Raemon's question).

Nod.

But, I've attempted that and mostly bounced off it because it felt too much like work. It might be the answer is "if you want to actually learn this thing you have to do the actual grownup thing" but by default I'm orienting around something like "the thing that seemed fun except that I didn't quite have enough resources to grok it, can I make it work better?" than "how do I seriously pursue learning math?"

(At least for the calculus videos I'm also skeptical about whether the "mastery is assumed" problem is especially bad, although I can imagine this being true for many of the more advanced stuff)

Some other sources of exercises you might want to check out (that have solutions and that I have used at least partly):

• Multiple choice quizzes (the ones related to linear algebra are determinants, elementary matrices, inner product spaces, linear algebra, linear systems, linear transformations, matrices, and vector spaces)
• Vipul Naik's quizzes (disclosure: I am friends with Vipul and also do contract work for him)

Regarding Axler's book (since it has been mentioned in this thread): there are several "levels" of linear algebra, and Axler's book is at a higher level (emphasis on abstract vector spaces and coordinate-free ways of doing things) than the 3Blue1Brown videos (more concrete, working in ). Axler's book also assumes that the reader has had exposure to the lower level material (e.g. he does not talk about row reduction and elementary matrices). So I'm not sure I would recommend it to someone starting out trying to learn the basics of linear algebra.

Gratuitous remarks:

• I think different resources covering material in a different order and using different terminology is in some sense a feature, not a bug, because it allows one to look at the subject from different perspectives. For instance, the "done right" in Axler's book comes from one such change in perspective.
• I find that learning mathematics well takes an unintuitively long time; it might be unrealistic to expect to learn the material well unless one puts in a lot of effort.
• I think there is a case to be made for the importance of struggling in learning (disclosure: I am the author of the page).

To add a little on different terminologies being a feature: if you ultimately want to apply linear algebra, you'll have to build a bridge from the theory to the particular application that is probably even more difficult than the bridge between two presentations of the theory. So it's probably good to practice building bridges.

(I'm also suspicious of people who say that the last book that they read on a subject was the best book, because that's when it clicked. How much did the first books prepare them?)

That does make sense.

If you want to get a feel of the mechanics, I recommend Gilbert Strang's Linear Algebra over Khan Academy and Brilliant.

I think part of the 3b1b series is based on that book and they have similar structures. Just sit down for 1-2 hours with some graph papers + mechanical pencil and do the end-of-chapter exercises, they are designed to reinforce the concepts.

Huh, interesting. I will check that out sometime.

I think you'll find any online exercises boring (too repetitive) or frustrating (too hard, with not enough clues with the exercises as to what you are not understanding). I think you need a person who knows the material and who is willing to trade skills with you.

I have a decent maths background, not stellar, and can have a go at an email exchange. In return what I would like from you 'as trade' is help from you with strategies for personal growth.

If you are interested in such a trade we can work out exactly how by emails. I'd imagine we'd be incrementally jointly creating small resources for those two areas of learning.

I can offer some linear algebra exercises around why a radar reflector works, and how eigen vectors matter in understanding the origins of life. We can also look at specific 3 blue 1 brown videos in that series and create new exercises directly around the content. 3 blue 1 brown is brilliant. I love what he is doing.

PM me if you are interested.

This is certainly optimal for me, assuming reasonable skill trade is possible. I need to figure out my schedule for the new year a bit but may reach out soonish about this.

It still feels like humanity is collectively leaving a sizeable chunk of value on the table by not having exercises that are reasonably tailored associated with the 3blue1brown content – obviously having a teacher is better than no teacher, but textbooks still exist and still strive to be relatively complete collections of concepts + opportunities to practice those concepts. I expect random person finding 3blue1brown on the internet to benefit a lot from having things to check their skill before continuing.

But the idea of working through this together and in the process creating some hopefully-longer-lasting resources sounds pretty good.