This is super nice. Seeing interactive graphics like this along with tutorial videos and the new Prism LaTeX editor from OpenAI make this an exciting time for math education. At the same time, AI advances on open problems in research and with LLM technology like with Axiom are making it an exciting time for math research as well.
Why are programmers always so attracted by these interactive/over-simplified/lightweight versions of linear algebra? They all focus on the visual aspects while ignoring the real stuff (theorems, proofs, etc.).
As a programmer who uses math heavily, I can answer this. As a programmer, your intuitive understanding of the world is infinitely important. You use it to formulate ideas, rule out solutions that would not be feasible, have an estimate of the expected cost and quality of solutions, etc.
Being able to dig deeper is important, but what's more important is to have an intuitive understanding of many many things: psychology, economy, finance, physics, art, etc. It's important to know the limits of your familiarity with any of these. For instance, my understanding of the fundamental practices in Accounting is really good (I've led a budget aggregation software for a huge conglomerate), but details is bad (tax rules for each industry, etc).
When I needed to create a software for optimizing stone cutting, I needed to know enough from computer vision, computational geometry, and optimization to know that our solution is feasible, task team members to learn what they need to do, and get into implementation, debugging and optimizing with them when needed.
After that, I still can't write code in computational geometry that handles all corner cases.
It's really good if we know everything with infinite precision, but for a programmer it's not efficient. We need to know where to stop.
In my experience, mathematicians are attracted by over-simplified/lightweight versions of programming. I think "just tell me what I need to do my job" is a universal human principle.
I don’t think they are simplified but they are an introduction course for sure. I think what’s usually missing is integrating (heh) linear algebra with calculus for solving more interesting problems but I think the audience for that needs to have a big appetite first. And those are hard to come by unless it’s pushed on you to finish your degree. And even then, people barely scrape by in those classes.
All this LLM stuff uses pretty basic linear algebra, and that’s a hot topic these days (not to turn my nose up at it, doing easy linear algebra at massive scales turned out to be a really good idea). Maybe that explains some of the attraction?
Because, realistically that's all programmers ever need, would be my guess. I do think linear algebra is an extremely interesting topic in its own right / outside, but yeah.
For the same reason mathematicians who aren't computer scientists or logicians are attracted to proof assistants implementing ZFC object languages while ignoring the stuff underneath (type theory, systems theory, etc.)
Maybe I should’ve been more clear: what I meant is that these programmers-oriented resources are all about applications…maybe a bit too much. For instance, In this book I cannot find the algebraic structure of vector spaces, theorems about how certain linear operators such as the kernel or the image led to subspaces (which is a very important result) or a proper introduction to the spectral theorem, for both Euclidean and Hermitian spaces (which also allows you to introduce some nice functional analysis).
What makes someone a "programmer" vs a mathematician? If you suspended your arrogance for a few moments you might realize you've wasted our time with a question that answers itself.
I mean it's also a generalization of very little value. Is someone doing linear algebra in lean not doing "the real stuff"? What is a programmer to you? Your question is why do some people not follow an area that is tangential to them to the maximal extent? Or to some arbitrary level of "real" as you subjectively define it? Is your claim that the level of linear algebra offered here is inherently useless unless paired with the "real stuff"?
Or did you just want us all to know you are a practitioner of such arts? Gold star for you.
Besides, I've never met a physicist or a (real) engineer who would go to such lengths to oversimplify the math part, even if it was only “tangential” to them.
Very nice. Clean presentation that tells you what you need to know to move from one section to the next, which is more than I can say for most of these efforts.
The 'tooltips' are also a nice touch. If someone wanted to go nuts with this concept, they could allow the user to highlight any sentence, equation, or individual symbol to bring up an 'Explain this' popup option.
This is super nice. Seeing interactive graphics like this along with tutorial videos and the new Prism LaTeX editor from OpenAI make this an exciting time for math education. At the same time, AI advances on open problems in research and with LLM technology like with Axiom are making it an exciting time for math research as well.
I LOVE this book! Where was it when I was learning algebra, years ago??
Thanks for posting. I wish there were many other books done similarly.
Selfishly, I'd love to see statistics, probability and advanced robotics displayed this way.
<3 <3
Well there is:
https://mathcs.clarku.edu/~djoyce/java/elements/elements.htm...
for geometry.
(previously I was noting a physics PDF set as well, but it's apparently not well-grounded/is controversial)
Making intuitive illustrations and graphs was very time consuming.
Now with LLMs it is so much easier and faster. Hopefully books will be rewritten.
Agreed on hoping books will be rewritten. It seems to be happening, but slowly. E.g., Harvard Machine Learning Systems: https://mlsysbook.ai/vol1/
lowkey wish I was in school again. this is so cool
You don't have to be in school to learn!
Why are programmers always so attracted by these interactive/over-simplified/lightweight versions of linear algebra? They all focus on the visual aspects while ignoring the real stuff (theorems, proofs, etc.).
As a programmer who uses math heavily, I can answer this. As a programmer, your intuitive understanding of the world is infinitely important. You use it to formulate ideas, rule out solutions that would not be feasible, have an estimate of the expected cost and quality of solutions, etc.
Being able to dig deeper is important, but what's more important is to have an intuitive understanding of many many things: psychology, economy, finance, physics, art, etc. It's important to know the limits of your familiarity with any of these. For instance, my understanding of the fundamental practices in Accounting is really good (I've led a budget aggregation software for a huge conglomerate), but details is bad (tax rules for each industry, etc).
When I needed to create a software for optimizing stone cutting, I needed to know enough from computer vision, computational geometry, and optimization to know that our solution is feasible, task team members to learn what they need to do, and get into implementation, debugging and optimizing with them when needed.
After that, I still can't write code in computational geometry that handles all corner cases.
It's really good if we know everything with infinite precision, but for a programmer it's not efficient. We need to know where to stop.
In my experience, mathematicians are attracted by over-simplified/lightweight versions of programming. I think "just tell me what I need to do my job" is a universal human principle.
I don’t think they are simplified but they are an introduction course for sure. I think what’s usually missing is integrating (heh) linear algebra with calculus for solving more interesting problems but I think the audience for that needs to have a big appetite first. And those are hard to come by unless it’s pushed on you to finish your degree. And even then, people barely scrape by in those classes.
I am kinda this person, any good resources/texts on this? I'm very slow but I have no deadlines.
All this LLM stuff uses pretty basic linear algebra, and that’s a hot topic these days (not to turn my nose up at it, doing easy linear algebra at massive scales turned out to be a really good idea). Maybe that explains some of the attraction?
Because, realistically that's all programmers ever need, would be my guess. I do think linear algebra is an extremely interesting topic in its own right / outside, but yeah.
For the same reason mathematicians who aren't computer scientists or logicians are attracted to proof assistants implementing ZFC object languages while ignoring the stuff underneath (type theory, systems theory, etc.)
Have you opened the book? There are both theorems and proofs there.
Maybe I should’ve been more clear: what I meant is that these programmers-oriented resources are all about applications…maybe a bit too much. For instance, In this book I cannot find the algebraic structure of vector spaces, theorems about how certain linear operators such as the kernel or the image led to subspaces (which is a very important result) or a proper introduction to the spectral theorem, for both Euclidean and Hermitian spaces (which also allows you to introduce some nice functional analysis).
Why do you consider a "real" textbook for linear algebra?
Can you read your own writing?
What makes someone a "programmer" vs a mathematician? If you suspended your arrogance for a few moments you might realize you've wasted our time with a question that answers itself.
I mean it's also a generalization of very little value. Is someone doing linear algebra in lean not doing "the real stuff"? What is a programmer to you? Your question is why do some people not follow an area that is tangential to them to the maximal extent? Or to some arbitrary level of "real" as you subjectively define it? Is your claim that the level of linear algebra offered here is inherently useless unless paired with the "real stuff"?
Or did you just want us all to know you are a practitioner of such arts? Gold star for you.
Bold of you to talk about arrogance…
Besides, I've never met a physicist or a (real) engineer who would go to such lengths to oversimplify the math part, even if it was only “tangential” to them.
Chill Winston.
Very nice. Clean presentation that tells you what you need to know to move from one section to the next, which is more than I can say for most of these efforts.
The 'tooltips' are also a nice touch. If someone wanted to go nuts with this concept, they could allow the user to highlight any sentence, equation, or individual symbol to bring up an 'Explain this' popup option.
dude i remember seeing this a long time ago from explorabl.es
never trusting something made by a swede (yes im norwegian)