4 ms·

Immersive Linear Algebra Book with Interactive Figures (2015)

295 points by srean 3mo ago

Cshaya 3mo ago
lowkey wish I was in school again. this is so cool
- spidersouris 3mo ago
  You don't have to be in school to learn!
aanet 3mo ago
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
- WillAdams 3mo ago
  Well there is: https://mathcs.clarku.edu/~djoyce/java/elements/elements.html 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)
CamperBob2 3mo ago
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.
whatever1 3mo ago
Making intuitive illustrations and graphs was very time consuming. Now with LLMs it is so much easier and faster. Hopefully books will be rewritten.
- akman 3mo ago
  Agreed on hoping books will be rewritten. It seems to be happening, but slowly. E.g., Harvard Machine Learning Systems: https://mlsysbook.ai/vol1/ https://mlsysbook.ai/vol1/
zaydiscool777 3mo ago
dude i remember seeing this a long time ago from explorabl.es
tjsquared 3mo ago
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.
deleted 3mo ago
[deleted]
RedCinnabar 3mo ago
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.).
- renyicircle 3mo ago
  Have you opened the book? There are both theorems and proofs there.
  - RedCinnabar 3mo ago
    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).
    - renyicircle 3mo ago
      I get where you're coming from but linear algebra is one of the most applied branches of mathematics out there. It's easy to visualize and easy to teach the computational side of it so it makes sense that some books focus more on that and these are popular among people whose main interest is in how to apply it - like programmers. You don't need to know what a ring is to learn how to multiply numbers. I wouldn't say this one is programmers-oriented either - it was posted to HN, sure, but maths and physics students could benefit from some visualizations too as a supplement to a more rigorous text. As for the things you mentioned as omitted, I think if you want to do a complete treatment you have to introduce more mathematical prerequisites which limits the audience of your book. For a mathematician, I'd say classical linear algebra in itself is not particularly interesting - it's very well-behaved and pretty much "solved". What you care about is how it relates to other structures (groups, modules, fields), how it develops into other topics (functional analysis) and how it's used to study other objects (representation theory, tangent spaces of manifolds). In isolation, most of what's left is the computational aspect which is what non-mathematicians mostly care about.
MariusGjerd 3mo ago
never trusting something made by a swede (yes im norwegian)
ShadeMe 3mo ago
[flagged]
zvmaz 3mo ago
3Blue1Brown[1] has opened my eyes on the intuitive understanding of linear algebra, many, many years after taking a first course in linear algebra, which left me completely mystified. [1] https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...
- apricot 3mo ago
  I teach linear algebra in college, and during our first class, I urge all my students to watch this series as soon as possible. It's great at giving them a geometrical understanding of what's going on, so they feel more motivated to learn the theorems and do computation (and already develop some intuition as to why the theorems are true).
  - sn9 3mo ago
    Have you read this post on incorporating spaced repetition into teaching linear algebra? https://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html https://bentilly.blogspot.com/2009/09/teaching-linear-algebr...
    - apricot 3mo ago
      I hadn't, thanks! Spaced repetition is certainly a useful tool. I always try to start every lecture with one or two review problems (that I call "brick problems", making sure the previous brick row is solid before building on it). I don't assign homework anymore because I have no interest in grading AI output (and even less interest in asking AI to grade AI output). The inverted classroom approach works well for me. Students read the textbook on their own (and I encourage them to use office hours, collaborate, or use AI tools if they need more explanations), doing checkpoint exercises to make sure they understand the material, and class time is spent on working problems.

295 points by srean 3mo ago

Cshaya 3mo ago

lowkey wish I was in school again. this is so cool

spidersouris 3mo ago
You don't have to be in school to learn!

aanet 3mo ago

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

WillAdams 3mo ago
Well there is: https://mathcs.clarku.edu/~djoyce/java/elements/elements.html 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)

CamperBob2 3mo ago

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.

whatever1 3mo ago

Making intuitive illustrations and graphs was very time consuming. Now with LLMs it is so much easier and faster. Hopefully books will be rewritten.

akman 3mo ago
Agreed on hoping books will be rewritten. It seems to be happening, but slowly. E.g., Harvard Machine Learning Systems: https://mlsysbook.ai/vol1/ https://mlsysbook.ai/vol1/

zaydiscool777 3mo ago

dude i remember seeing this a long time ago from explorabl.es

tjsquared 3mo ago

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.

deleted 3mo ago

[deleted]

RedCinnabar 3mo ago

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.).

renyicircle 3mo ago
Have you opened the book? There are both theorems and proofs there.
- RedCinnabar 3mo ago
  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).
  - renyicircle 3mo ago
    I get where you're coming from but linear algebra is one of the most applied branches of mathematics out there. It's easy to visualize and easy to teach the computational side of it so it makes sense that some books focus more on that and these are popular among people whose main interest is in how to apply it - like programmers. You don't need to know what a ring is to learn how to multiply numbers. I wouldn't say this one is programmers-oriented either - it was posted to HN, sure, but maths and physics students could benefit from some visualizations too as a supplement to a more rigorous text. As for the things you mentioned as omitted, I think if you want to do a complete treatment you have to introduce more mathematical prerequisites which limits the audience of your book. For a mathematician, I'd say classical linear algebra in itself is not particularly interesting - it's very well-behaved and pretty much "solved". What you care about is how it relates to other structures (groups, modules, fields), how it develops into other topics (functional analysis) and how it's used to study other objects (representation theory, tangent spaces of manifolds). In isolation, most of what's left is the computational aspect which is what non-mathematicians mostly care about.

MariusGjerd 3mo ago

never trusting something made by a swede (yes im norwegian)

ShadeMe 3mo ago

[flagged]

zvmaz 3mo ago

3Blue1Brown[1] has opened my eyes on the intuitive understanding of linear algebra, many, many years after taking a first course in linear algebra, which left me completely mystified. [1] https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...

apricot 3mo ago
I teach linear algebra in college, and during our first class, I urge all my students to watch this series as soon as possible. It's great at giving them a geometrical understanding of what's going on, so they feel more motivated to learn the theorems and do computation (and already develop some intuition as to why the theorems are true).
- sn9 3mo ago
  Have you read this post on incorporating spaced repetition into teaching linear algebra? https://bentilly.blogspot.com/2009/09/teaching-linear-algebra.html https://bentilly.blogspot.com/2009/09/teaching-linear-algebr...
  - apricot 3mo ago
    I hadn't, thanks! Spaced repetition is certainly a useful tool. I always try to start every lecture with one or two review problems (that I call "brick problems", making sure the previous brick row is solid before building on it). I don't assign homework anymore because I have no interest in grading AI output (and even less interest in asking AI to grade AI output). The inverted classroom approach works well for me. Students read the textbook on their own (and I encourage them to use office hours, collaborate, or use AI tools if they need more explanations), doing checkpoint exercises to make sure they understand the material, and class time is spent on working problems.