Why not... show it visually, manipulate it visually, explain in layman's terms and THEN explain the formal properties? That is the approach 3Blue1Brown takes to great success.
So, definitely useful renderings, but I'm not sure it's any more "intuitive"* than the mainstream approaches.
* In quotes to indicate that there is quite a bit of subjectivity in that term.
The explanations are definitely very lacking of formalisms. There even isn't any proper explanation of what a vector space actually is.
It doesn't seem to pretend to be a full textbook replacement, more like a supplementary material, so I don't think it matters all that much.
I really like the visual representation of elementary row operations, but the intersection of the three planes is off their edges, so isn't actually shown. Also, "any vector multiplied by the three standard basis vectors in matrix form, i, j, k" uses a really complicated strange way to say "identity matrix" - you're not actually talking about multiplying by the standard basis vectors.
I found a bug (Chrome 65.0.3325.181, Windows): The scroll position is kept when you switch sections, so you have to manually scroll up to the top.
Great feedback! I had been thinking of doing something like this but I was waiting on actual readers to mention it. I'll see if I can get to it this weekend.
> I really like the visual representation of elementary row operations, but the intersection of the three planes is off their edges, so isn't actually shown.
Yeah, good point. In the last visualization we have a vector pointing to the intersection, perhaps a similar sort of thing can be leveraged there.
> I found a bug (Chrome 65.0.3325.181, Windows): The scroll position is kept when you switch sections, so you have to manually scroll up to the top.
Thanks for the report! I think someone else mentioned this already and I filed a ticket on the github repository.
(These are some of the finest educational materials for mathematics that have ever been created in any media, they really are that good.)
The gap which I'm trying to fill here is that 3b1b goes over the higher level intuitions but tends to skip over some of the details at times (which is fair enough, they're a pain to demonstrate through video and animation). The idea I had here was to have worked examples as well.
Different tastes for different people. I found this website to be more useful, since I can browse, reread and skim through the text much more easily/faster than on video.
1. There are a few grammatical errors (missing word, missing comma) that are forgivable but I find that any imprecision in language is a big detriment for a student trying to understand an explanation. I tend to get hung up on those since, not knowing the material, I can't resolve ambiguities with context. Since subtleties are important in math it would be worthwhile to go over the explanations again with a fine-toothed comb to ensure there are no typos or amphiboly.
2. The visuals aren't always clear. For example, volumes don't show edges so it's hard to make out that you're seeing a volume rather than an irregular area (and which volume you're seeing). Also I think a caption to each image could be useful, to describe how the shape demonstrates the concept. Something like "this visual represents a shear transformation of the matrix described above", just to be really explicit and clear.
> There are a few grammatical errors (missing word, missing comma) that are forgivable but I find that any imprecision in language is a big detriment for a student trying to understand an explanation. I tend to get hung up on those since, not knowing the material, I can't resolve ambiguities with context. Since subtleties are important in math it would be worthwhile to go over the explanations again with a fine-toothed comb to ensure there are no typos or amphiboly.
Yeah, this is a pet peeve of mine for sure. I am not very precise when it comes to written prose.
I was thinking of airtasking out a grammar check, since I'm pretty awful at catching my own mistakes here. I might just do that.
> The visuals aren't always clear. For example, volumes don't show edges so it's hard to make out that you're seeing a volume rather than an irregular area (and which volume you're seeing). Also I think a caption to each image could be useful, to describe how the shape demonstrates the concept. Something like "this visual represents a shear transformation of the matrix described above", just to be really explicit and clear.
Great idea. I've added it to my list at https://github.com/smspillaz/intuitive-math/issues/8
The "voice"/style of the writing has a great balance between giving theoretical information and keeping it comprehensible without having to read like 5 times (which most algebra texts imo really botch!).
Most algebra concepts (in the linear/modern/abstract sense) have an intuitive/mental-model foundation, often graphical. It's disappointing that most texts don't explore the intuition in a friendly way like this text does. (The theory is deeper than the intuition and intuition can be very wrong at times, but anyone studying advanced-ish math like this is capable of realizing that!)
One annoyance is that the graphs don't render if they're slightly off-screen (and they render slowly/fade-in or something) so it can be somewhat annoying to scroll back/forth between graphic examples.
> One annoyance is that the graphs don't render if they're slightly off-screen (and they render slowly/fade-in or something) so it can be somewhat annoying to scroll back/forth between graphic examples.
Yeah. This was an engineering tradeoff - there's a limit on the number of WebGL contexts (15 for Firefox, for instance) that can be running at any time so I had to turn off rendering of the visualizations when offscreen.
If anyone knows a better way to approach this, please let me know!
[0] http://www.gabrielgambetta.com/computer-graphics-from-scratc...
I'm receiving lots of great links in the comments here, so it'd be great to cross link similar resources on this page too.
