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Immersive Linear Algebra – textbook with fully interactive figures (2015)

1015 points by sebg 10y ago

ydmitry 10y ago
Try use web workers to not block UI while processing: http://immersivemath.com/ila/ch05_gausselim/ch05.html http://immersivemath.com/ila/ch05_gausselim/ch05.html
aligajani 10y ago
Wow, this is great.
- francopellicie 10y ago
  Al right, where do we get it?
bobajeff 10y ago
I was just wondering recently why something like this didn't exist when trying to learn about Matrixes via Kahn Academy.
yellow_postit 10y ago
One thing missing in comparison to most other textbooks is a set of problems to test the reader's understanding. The rotating and interactive figures are a very nice touch though. Along the lines of interactivity, maybe having a scratch area like a Jupyter notebook would be a potentially great addition so that I could try problems near the area where I'm reading.
- happy-go-lucky 10y ago
  I thought the same thing. A set of problems to test the reader's understanding and a scratch area like a Jupyter notebook would improve the interactivity.
  - hardcore96 10y ago
    I've recently worked on http://dspillustrations.com http://dspillustrations.com for a pictorial description of signal processing concepts. In the online version, it's not fully interactive, just animations. But after downloading, you can certainly change formulas and run it interactively. Would be interested how one would manage to include a jupyter scratchpad in the online version?
    - happy-go-lucky 10y ago
      Embedding Jupyter Widgets in Other Contexts than the Notebook: http://ipywidgets.readthedocs.io/en/latest/embedding.html http://ipywidgets.readthedocs.io/en/latest/embedding.html Temporary notebook service: https://github.com/jupyter/tmpnb https://github.com/jupyter/tmpnb Jupyter javascript plugin for static sites: https://github.com/oreillymedia/thebe https://github.com/oreillymedia/thebe Also, you may want to consider trinket: https://trinket.io/ https://trinket.io/ http://www.hnwatcher.com/r/984396/Embed-interactive-Python-anywhere-on-the-web http://www.hnwatcher.com/r/984396/Embed-interactive-Python-a... For example, this book uses trinket for interactive Python: https://books.trinket.io/pfe/index.html https://books.trinket.io/pfe/index.html I've just tried running: import numpy as np from scipy import linalg A = np.array([[1,2],[3,4]]) print(linalg.det(A)) at https://trinket.io/features/python3 https://trinket.io/features/python3 and it worked like a charm :)
      - hardcore96 10y ago
        Thanks for these great references! I will definitely have a look at them and see if I can use them for my purposes.
aarongeisler 10y ago
I really like this. This would've been handy as a student. I definitely prefer this to a textbook.
tomrod 10y ago
This is fantastic, and where I have expected e-textbooks to go for years. Kudos!!!
blt 10y ago
So glad someone made this. I'm a "true believer" in animations and interactivity for learning math. They can make some concepts instantly intuitive that take a while to grasp symbolically.
Dangeranger 10y ago
This is very nice. As a suggestion for improvement, consider allowing the learner to edit the formulas which represent the figures and have the figures update. Additionally editing the figures could update the formula in real time. This sort of bi-directional instant feedback will aid the understanding and engagement of the learner better than figure manipulation alone.
- problems 10y ago
  Sounds like a good idea - one of the main ways I learned programming... and still wind up doing all the time is to just copy examples and hack them into shape, understanding what's necessary, what's not and how they'll respond to different parameters via experimentation.
laretluval 10y ago
If you're interested in more great animations for a visual understanding of linear algebra, I can't recommend 3Blue1Brown's "Essence of Linear Algebra" series on Youtube highly enough. https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
- teperpencoli 10y ago
  These are the best linear algebra videos I've ever seen. The creator is now working on an introductory calculus series. He's already released the first few episodes to patrons.
  - mrkgnao 10y ago
    Wasn't he hired by Khan Academy recently? It's great if that helps a wider audience to benefit from his work.
    - ludicast 10y ago
      I think it's the reverse. He did all or most of their multivariable calc series, and maybe some other things for them. And then started to do his own thing a little more.
- metaobject 10y ago
  Holy shit. I wish I'd had access to these videos during my Mattix Algebra (I think it was called) course in undergrad. I feel like my lack of truly understanding LA has somewhat impaired my grasp of some of the other topics covered in CS undergrad and grad school.
  - Berobero 10y ago
    Seriously... the hallmark of all the linear algebra classes I had was the instructor completely failing to demonstrate the various concepts in visual terms. And that's pretty unforgivable, since linear algebra feels like one of those subjects where, until you have some key components of it modeled in your head that serve to glue everything together, every new may as well be random words pulled from a foreign dictionary.
ktta 10y ago
Although the focus with this book is with the fully interactive figures, I'm more impressed that ALL of the book is available for free. I'm constantly seeing more people coming out with books for free or just a passive donation link. This makes me immensely happy seeing as how they're leveraging the available free resources (Latex, CC-BY-SA content, free software for graphics) to make more resources available for free. Open software is one thing, but a book is much more permanent in my opinion. A book like this will never go 'stale' or old like software does. We only need handful of good books for every topic out there at which point we can basically not buy books anymore. For many topics, I hardly have to consider buying a book since I can just use a free book offered by a professor. And watch course lectures. What I want to say is this: Please write more for free. It doesn't matter if there is not much interest in what you are writing. It will help you too!
- a3n 10y ago
  > Although the focus with this book is with the fully interactive figures, I'm more impressed that ALL of the book is available for free. That was exactly my first thought. "Wait a minute, this is an entire textbook, with all the work, and the prior experience, hiding in plain site behind cool interactivity."
- swiley 10y ago
  Well written software doesn't seem to go stale, or at least not quickly and poorly written books do.
  - ktta 10y ago
    Well, now that I think about it, the software going 'stale' depends on the domain. You can write a networking stack in C adhering to a protocol and be done with it, while code for the new $COOL_WEB_APP might go out of mainstream use by the time you're done writing the networking stack. I don't think most books, especially math and science ones go stale. A poorly written book is just that; a poorly written book. It's bad from the beginning
mrkgnao 10y ago
Excellent work! I've been trying to get my mother (she's a physics teacher) to learn linear algebra properly for a long time. Artin didn't work (ha), Khan Academy moved too slow/bored her, but she seems interested in this. It's important to appreciate how useful it might be to make math "tangible". Sure, someone who can define a manifold by saying "oh, put charts on it, locally diffeo blah blah" probably has a good set of mental models that help them find analogies and even "tangibilize" (word?) new ideas. Once you learn the way abstraction works, broadly speaking, you can take the training wheels off: but lots of people never get past that stage. On one hand, I see a lot of people on HN talk about how the complicated notation of academic math/CS keep people out (and there is an understandable amount of resentment at people keeping "outsiders" out with this), and on the other hand, I sort of reflexively bristle (it's gotten lesser now) at people integrating the notion of an inner product into a vector space, because it is important to not stumble later when you find out your basic intuition for something is broken[3]. (Of course, intuition can be incrementally "patched": Terence Tao's essay[3] talks about this, from the perspective of someone who is a brilliant educator in addition to also being one of the most versatile mathematicians around.) Maybe presentations of basic mathematics that are - simple - rigorous - free of half-truths can be made accessible by using such visualizations and interactive techniques to decrease the perceived unfamiliarity of the ideas? I don't think there are many[2] treatments of mathematical topics that satisfy these criteria and yet manage to be approachable: one either skimps on a clean presentation (Khan Academy), or assumes a lot of mathematical maturity (shoutout to Aluffi!) from the reader. "Manipulable resources" might help fill this gap. It's an exciting time! -- In the section where they give examples of matrix inverses, to give people a sense of how important multiplication order is, they give an example of RHR'H' (using a prime for inverse, R for a rotation matrix, and H for a shear matrix). One of the most beautiful illustrations[1] in the book follows, with the four corners of a square moving independently in circles, and then the book states that It is quite close, but it is not at all useful. While I understand the need to clarify the importance of multiplying matrices in the correct order, maybe a short aside on the unreasonable (practical!) effectiveness of commutators[0] would be useful? [0]: https://en.wikipedia.org/wiki/Commutator https://en.wikipedia.org/wiki/Commutator [1]: http://immersivemath.com/ila/ch06_matrices/ch06.html http://immersivemath.com/ila/ch06_matrices/ch06.html ("Example 6.12: Matrix Product Inverse au Faux") [2]: Visual Complex Analysis is brilliant, though. [3]: https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
dang 10y ago
Discussed in 2015: https://news.ycombinator.com/item?id=10183725 https://news.ycombinator.com/item?id=10183725.
- idclip 10y ago
  I'm glad it's been reposted - writing an ana exam in 4 weeks!
freakynit 10y ago
Wow...i so so so much wanted a book just like this... Thanks
mkl 10y ago
Does anyone know more about how the figures were developed? I found https://www.lth.se/fileadmin/lth/genombrottet/LUkonf2015/41_Stro__m_etal.pdf https://www.lth.se/fileadmin/lth/genombrottet/LUkonf2015/41_... which says "The figures were programmed using JavaScript using a graphics engine that we have developed for the interactive illustrations", but that's all the detail it gives.
- immersivemath 10y ago
  Is there anything in particular that you would like to know?
kgarten 10y ago
on a tangent, I was just reading https://medium.com/@dominikus/the-end-of-interactive-visualizations-52c585dcafcb https://medium.com/@dominikus/the-end-of-interactive-visuali... in short (over simplified, my take): interactive visualizations are dead because nobody interacts with them. wondering if this holds here as well. Can't find the reference anymore, but there were also papers in educational sciences that interactive books usually don't increase comprehension of kids (they just play with them instead of depen their understanding). edit: sorry didn't want to sound too critical. The work is awesome (upvote), was just thinking out loud.
- tbihl 10y ago
  For me this gets to a deeper point. While there are probably some people who innately understand tough mathematical or scientific concepts, it's been my experience that learning is largely about spending enough time struggling diligently with the material until you've built a robust and correct mental model of it. Then on to exercises to see if your model breaks or holds up when strained. If you approach interactive aids and videos and images with this mindset, they can be very helpful. I can also clearly remember times when this was not my approach, and in those times, pictures were just a lot of time saved because I didn't have to read text that otherwise would have been there.
- jimhefferon 10y ago
  Thank you for the link. I learned from it. I am the author of a widely-used LA text, and have considered adding interactive stuff. But there is a tradeoff. For one thing, it locks you to online, and despite the claims of our IT people, my correspondents (mostly self-learners) do not want online, they want print or PDF (as do I, since the appearance that LaTeX gives me is important to me). For another, the tech has in the recent past changed so fast that maintaining the interactives would be a significant job. I don't mind learning JS to do something good but tying myself to many hours a year responding to bug reports from people on obscure platforms, or using IE6, is not a good use of my life energy. Finally, I had a colleague, a complex analyst, try Visual Complex Analysis and he reported that students did not get it. He is very sober, very caring, very reliable. This starts to make sense of his report.
markbao 10y ago
Amazing resource, but is it still being updated? Last update seems to be from back in July last year.
- immersivemath 10y ago
  New chapter out today! This time on linear mappings.
eellpp 10y ago
Last year, i saw the videos by 3Blue1Brown and inspired by it went on to read some of the standard text books like linear algebra application by Strang. I had seen the Strang videos earlier but somehow did not follow it through. This time however my perspective was changed. I was approaching the subject through the lens of intuition and simplicity. Wherever i found something challenging, i waited for the next day to again re-do it (because i know that this should not be that complex to understand or the understanding is wrong). And to my surprise, again and again, the difficulty was in my rigidness in understanding. The next day , or even later during the day, when my mind was fresh again, i can reason through the concept and get the intuition behind it. Since then i have seen the Strang videos again and again. Beginning to end. Read the book chapter by chapter and exercises by exercise. And what a delight it had been. And then i jumped upon Joe Blitzstein's probability lectures. What a blast ! Is there a list of teachers like these there, who in the pretext of teaching algebra/probability etc are in reality wiring up our thinking process in ways immaterial to subject they are teaching. Many of us don't want the material to be too casual/layman terms (which hampers self understanding as its no challenging anything within us) and not too rigid (where we cannot break through the challenge).
- j2kun 10y ago
  > the difficulty was in my rigidness in understanding At what point did you realize this? Like, could you provide a specific example of a topic you thought was hard at first but later came back to and realized was all about the intuition?
  - Retra 10y ago
    Not the OP, but... I was just reading Landau & Lifshitz' "Statistical Physics", and can reflect on a series of thoughts that may elaborate on how intuition plays a role in the enjoyment and understanding of complex material. I've been meaning to write it down anyway... On page 3, the book says "A fundamental feature of this [closed system/open subsystem] approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state." When I first read this, I thought "non sequitur, but whatever, I'll continue..." Now, the context of this quote is that the authors are trying to explain why statistical methods work at all. And they said prior that we start with laws that apply to 'microscopic' particles and use statistics to generalize to 'macroscopic' systems. However, the second time I read this, I kept thinking it must be backwards. We didn't understand the motion of (classical) protons before understanding the motion of macroscopic balls. So we had to have been operating under the assumption that the macroscopic laws must apply to microscopic objects, and then require that the must also be reproducible macroscopically through statistical methods. That is, we require that these laws be invariant across the microscopic-to-macroscopic transition. But to do that, we have to use a framework which expresses such a transition. So, for instance, if we are reasoning about the motion of a ball, we have to translate our laws into laws over the motion of some statistical model of the ball. Say, it's center of mass. And with this concept in hand, we could write laws that apply to both the macro and micro worlds, since a 'center of mass' is a macro-micro-scale-invariant abstraction. So we partition the space of all possible laws of nature, and chose to work only in that partition which encodes things we can actually know about nature -- the partition identified by the macro-micro-scale-invariant. So re-reading that passage, it is now not a non-sequitur for me. Now it says "because the interactions with the outside world are so complex, we could not hope to predict their influence. Thus we are justified in using random variables to model their influence, and concepts that derive from the use of random variables to ensure we have complexity-scale invariance when we formulate our laws of physics." And this is not a non-sequitur to me. It follows directly from the meaning of the word "random." Of course statistical methods work when the complexity of a system is indistinguishable from randomness. -- And this whole line of thought generalizes (albeit informally.) For instance, the meaning of words is an invariant across a long thread of contextual translations, and these invariants are used the same way: to partition the space of all possible meanings in such a way that one partition contains all of the 'knowledge' imbued by that word, and thus you can navigate a narrower space of meaning to find the intended and/or correct one. Gives me a certain brand of appreciation for good poetry. Or -- my girlfriend -- who recently told me that she loved algebra but couldn't understand trigonometry. I tried telling her that the algebraic transformations were invariant-preserving operations selected because they conformed to known laws about 'functions' like addition and multiplication which have commutativity and identity laws, and that trigonometry was no different: different functions, but all of the algebraic transformations you needed were selected from the laws of trigonometry with the purpose of maintaining the exact same invariants. (Not that she cared much, to be honest...) And on and on... I could probably talk for days about all the different ways every subject can be reduced to transformations and invariants and how they are used to solve problems.
digitalshankar 10y ago
Can anyone hack this and make a PDF book?
- krackers 10y ago
  Doesn't that defeat the point of intractability?
  - benelot 10y ago
    Interactivity maybe, intractability is not having the property of being tractable. For an algorithm, this means that its performance can not be described in polynomials. But your point is still valid, a pdf does not support animations, thus a pdf would be useless.
    - digitalshankar 10y ago
      I didn't ask PDF for interactivity, i asked PDF for taking notes. The theory is so solid.
ErikBjare 10y ago
I'm a MSc in CS student at the university where the authors (whom I've met a few times) teach. If you have any questions you think I'd know the answer to, ask away!
- amelius 10y ago
  Perhaps you can let them know their work is being discussed on HN :)
  - immersivemath 10y ago
    A good idea! Although now we are already aware. If you have any questions you can ask us here or at immersivemath@gmail.com
techman9 10y ago
I feel as if I'm looking at the future of textbooks. This is absolutely incredible.
darkhorn 10y ago
The order of the words and suffices is like in Turkish. One of my English teachers who has lived in Japan for a time (and she was a Turk) told us that it is very easy to learn Japanese because it follows the same word orders, and vica verse. On the other hand English is very hard becuese there is zero connection.
boramalper 10y ago
Reminded me of Explorable Explanations: http://explorableexplanations.com http://explorableexplanations.com
kowdermeister 10y ago
Jaw dropped. This is exactly what I needed to get some progress with my shader learning :)
- corysama 10y ago
  You might also like https://thebookofshaders.com https://thebookofshaders.com
proudofyou 10y ago
Maybe I'll try teaching this to my 11 year old sister. Wait, how can I teach calculus to an untalented 11 year old?
francopellicie 10y ago
Al right, where do we get it?
NumberCruncher 10y ago
I wish we had resources like this when I was on the university.
machiaweliczny 10y ago
This looks great! Could some of you share other nice books and video series covering core CS math basics?
blinry 10y ago
If you like this, you might be interested in /r/explorables, a subreddit filled with interactive stuff: https://www.reddit.com/r/explorables/ https://www.reddit.com/r/explorables/
Demcox 10y ago
This will come in handy for my LinAlg exam :3
BooglyWoo 10y ago
Mandelbrot cites his ability to "think in pictures" as fundamental to his process and insights. He offers some interesting reflections and anecdotes on this subject in this interview describing his classes préparatoires aux grandes écoles. Apparently his teacher considered him a total wildcard case who would either flunk the exams or pass with flying colours, because of his habit of approaching everything through geometric intuition rather than symbolic manipulation. https://www.webofstories.com/play/benoit.mandelbrot/6 https://www.webofstories.com/play/benoit.mandelbrot/6
mathgenius 10y ago
To me this book looks like a whole bunch of equations, with some fancy graphics sprinkled on top. And, far too many equations! Linear algebra is much more elegant (simple) than this. To pick one example, they define the inner product using the cosine of the "smallest angle between the two vectors." Sure if you want to calculate a number (an inner product in two dimensions), and you happened to know the angle in question, this might be helpful. But otherwise it completely obscures everything else about the inner product. How does an interactive graphic help you understand wtf is a cosine doing in this equation? What is a cosine anyway? Where is the graphic for that? This just seems too backward and over-done to me. But go ahead and test it on some newbies, maybe I'm totally wrong here.
- jacobolus 10y ago
  More importantly, the definition of the cosine is given by projecting a unit vector onto a given directed line, and then measuring the length of the projection. The way to find a given cosine of a given angle between two arbitrary vectors is by taking the dot product of the two vectors and then normalizing by their lengths. Using the cosine to define the dot product is precisely backwards.
sp4ke 10y ago
You might be interested by this curated list of explorable explanations [1]. It has a good section on math. [1] https://github.com/sp4ke/awesome-explorables https://github.com/sp4ke/awesome-explorables
jarek83 10y ago
Great content, makes it much easier to get through concepts. But as most math learning resources, it lacks examples of their practical application. "The law of cosines is a very useful formula to know" - fine but why? Most learners can't imagine where they can apply what their teachers say, which brings it down to learn useless-for-them terms, needed just to pass exams. Some basic examples like these would make it much more reasonable to learn: http://study.com/academy/lesson/solving-real-world-problems-using-the-law-of-cosines.html http://study.com/academy/lesson/solving-real-world-problems-...
msaharia 10y ago
Q: How would one go about creating such animations? What's the best workflow? Found the figures were slow to load. Great resource, though!

1015 points by sebg 10y ago

Try use web workers to not block UI while processing: http://immersivemath.com/ila/ch05_gausselim/ch05.html http://immersivemath.com/ila/ch05_gausselim/ch05.html

aligajani 10y ago

Wow, this is great.

francopellicie 10y ago
Al right, where do we get it?

bobajeff 10y ago

I was just wondering recently why something like this didn't exist when trying to learn about Matrixes via Kahn Academy.

yellow_postit 10y ago

One thing missing in comparison to most other textbooks is a set of problems to test the reader's understanding. The rotating and interactive figures are a very nice touch though. Along the lines of interactivity, maybe having a scratch area like a Jupyter notebook would be a potentially great addition so that I could try problems near the area where I'm reading.

happy-go-lucky 10y ago
I thought the same thing. A set of problems to test the reader's understanding and a scratch area like a Jupyter notebook would improve the interactivity.
- hardcore96 10y ago
  I've recently worked on http://dspillustrations.com http://dspillustrations.com for a pictorial description of signal processing concepts. In the online version, it's not fully interactive, just animations. But after downloading, you can certainly change formulas and run it interactively. Would be interested how one would manage to include a jupyter scratchpad in the online version?
  - happy-go-lucky 10y ago
    Embedding Jupyter Widgets in Other Contexts than the Notebook: http://ipywidgets.readthedocs.io/en/latest/embedding.html http://ipywidgets.readthedocs.io/en/latest/embedding.html Temporary notebook service: https://github.com/jupyter/tmpnb https://github.com/jupyter/tmpnb Jupyter javascript plugin for static sites: https://github.com/oreillymedia/thebe https://github.com/oreillymedia/thebe Also, you may want to consider trinket: https://trinket.io/ https://trinket.io/ http://www.hnwatcher.com/r/984396/Embed-interactive-Python-anywhere-on-the-web http://www.hnwatcher.com/r/984396/Embed-interactive-Python-a... For example, this book uses trinket for interactive Python: https://books.trinket.io/pfe/index.html https://books.trinket.io/pfe/index.html I've just tried running: import numpy as np from scipy import linalg A = np.array([[1,2],[3,4]]) print(linalg.det(A)) at https://trinket.io/features/python3 https://trinket.io/features/python3 and it worked like a charm :)
    - hardcore96 10y ago
      Thanks for these great references! I will definitely have a look at them and see if I can use them for my purposes.

aarongeisler 10y ago

I really like this. This would've been handy as a student. I definitely prefer this to a textbook.

tomrod 10y ago

This is fantastic, and where I have expected e-textbooks to go for years. Kudos!!!

blt 10y ago

So glad someone made this. I'm a "true believer" in animations and interactivity for learning math. They can make some concepts instantly intuitive that take a while to grasp symbolically.

Dangeranger 10y ago

This is very nice. As a suggestion for improvement, consider allowing the learner to edit the formulas which represent the figures and have the figures update. Additionally editing the figures could update the formula in real time. This sort of bi-directional instant feedback will aid the understanding and engagement of the learner better than figure manipulation alone.

problems 10y ago
Sounds like a good idea - one of the main ways I learned programming... and still wind up doing all the time is to just copy examples and hack them into shape, understanding what's necessary, what's not and how they'll respond to different parameters via experimentation.

laretluval 10y ago

If you're interested in more great animations for a visual understanding of linear algebra, I can't recommend 3Blue1Brown's "Essence of Linear Algebra" series on Youtube highly enough. https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

teperpencoli 10y ago
These are the best linear algebra videos I've ever seen. The creator is now working on an introductory calculus series. He's already released the first few episodes to patrons.
- mrkgnao 10y ago
  Wasn't he hired by Khan Academy recently? It's great if that helps a wider audience to benefit from his work.
  - ludicast 10y ago
    I think it's the reverse. He did all or most of their multivariable calc series, and maybe some other things for them. And then started to do his own thing a little more.
metaobject 10y ago
Holy shit. I wish I'd had access to these videos during my Mattix Algebra (I think it was called) course in undergrad. I feel like my lack of truly understanding LA has somewhat impaired my grasp of some of the other topics covered in CS undergrad and grad school.
- Berobero 10y ago
  Seriously... the hallmark of all the linear algebra classes I had was the instructor completely failing to demonstrate the various concepts in visual terms. And that's pretty unforgivable, since linear algebra feels like one of those subjects where, until you have some key components of it modeled in your head that serve to glue everything together, every new may as well be random words pulled from a foreign dictionary.

ktta 10y ago

Although the focus with this book is with the fully interactive figures, I'm more impressed that ALL of the book is available for free. I'm constantly seeing more people coming out with books for free or just a passive donation link. This makes me immensely happy seeing as how they're leveraging the available free resources (Latex, CC-BY-SA content, free software for graphics) to make more resources available for free. Open software is one thing, but a book is much more permanent in my opinion. A book like this will never go 'stale' or old like software does. We only need handful of good books for every topic out there at which point we can basically not buy books anymore. For many topics, I hardly have to consider buying a book since I can just use a free book offered by a professor. And watch course lectures. What I want to say is this: Please write more for free. It doesn't matter if there is not much interest in what you are writing. It will help you too!

a3n 10y ago
> Although the focus with this book is with the fully interactive figures, I'm more impressed that ALL of the book is available for free. That was exactly my first thought. "Wait a minute, this is an entire textbook, with all the work, and the prior experience, hiding in plain site behind cool interactivity."
swiley 10y ago
Well written software doesn't seem to go stale, or at least not quickly and poorly written books do.
- ktta 10y ago
  Well, now that I think about it, the software going 'stale' depends on the domain. You can write a networking stack in C adhering to a protocol and be done with it, while code for the new $COOL_WEB_APP might go out of mainstream use by the time you're done writing the networking stack. I don't think most books, especially math and science ones go stale. A poorly written book is just that; a poorly written book. It's bad from the beginning

mrkgnao 10y ago

Excellent work! I've been trying to get my mother (she's a physics teacher) to learn linear algebra properly for a long time. Artin didn't work (ha), Khan Academy moved too slow/bored her, but she seems interested in this. It's important to appreciate how useful it might be to make math "tangible". Sure, someone who can define a manifold by saying "oh, put charts on it, locally diffeo blah blah" probably has a good set of mental models that help them find analogies and even "tangibilize" (word?) new ideas. Once you learn the way abstraction works, broadly speaking, you can take the training wheels off: but lots of people never get past that stage. On one hand, I see a lot of people on HN talk about how the complicated notation of academic math/CS keep people out (and there is an understandable amount of resentment at people keeping "outsiders" out with this), and on the other hand, I sort of reflexively bristle (it's gotten lesser now) at people integrating the notion of an inner product into a vector space, because it is important to not stumble later when you find out your basic intuition for something is broken[3]. (Of course, intuition can be incrementally "patched": Terence Tao's essay[3] talks about this, from the perspective of someone who is a brilliant educator in addition to also being one of the most versatile mathematicians around.) Maybe presentations of basic mathematics that are - simple - rigorous - free of half-truths can be made accessible by using such visualizations and interactive techniques to decrease the perceived unfamiliarity of the ideas? I don't think there are many[2] treatments of mathematical topics that satisfy these criteria and yet manage to be approachable: one either skimps on a clean presentation (Khan Academy), or assumes a lot of mathematical maturity (shoutout to Aluffi!) from the reader. "Manipulable resources" might help fill this gap. It's an exciting time! -- In the section where they give examples of matrix inverses, to give people a sense of how important multiplication order is, they give an example of RHR'H' (using a prime for inverse, R for a rotation matrix, and H for a shear matrix). One of the most beautiful illustrations[1] in the book follows, with the four corners of a square moving independently in circles, and then the book states that It is quite close, but it is not at all useful. While I understand the need to clarify the importance of multiplying matrices in the correct order, maybe a short aside on the unreasonable (practical!) effectiveness of commutators[0] would be useful? [0]: https://en.wikipedia.org/wiki/Commutator https://en.wikipedia.org/wiki/Commutator [1]: http://immersivemath.com/ila/ch06_matrices/ch06.html http://immersivemath.com/ila/ch06_matrices/ch06.html ("Example 6.12: Matrix Product Inverse au Faux") [2]: Visual Complex Analysis is brilliant, though. [3]: https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/there%E2%80%99s...

dang 10y ago

Discussed in 2015: https://news.ycombinator.com/item?id=10183725 https://news.ycombinator.com/item?id=10183725.

idclip 10y ago
I'm glad it's been reposted - writing an ana exam in 4 weeks!

freakynit 10y ago

Wow...i so so so much wanted a book just like this... Thanks

mkl 10y ago

Does anyone know more about how the figures were developed? I found https://www.lth.se/fileadmin/lth/genombrottet/LUkonf2015/41_Stro__m_etal.pdf https://www.lth.se/fileadmin/lth/genombrottet/LUkonf2015/41_... which says "The figures were programmed using JavaScript using a graphics engine that we have developed for the interactive illustrations", but that's all the detail it gives.

immersivemath 10y ago
Is there anything in particular that you would like to know?

kgarten 10y ago

on a tangent, I was just reading https://medium.com/@dominikus/the-end-of-interactive-visualizations-52c585dcafcb https://medium.com/@dominikus/the-end-of-interactive-visuali... in short (over simplified, my take): interactive visualizations are dead because nobody interacts with them. wondering if this holds here as well. Can't find the reference anymore, but there were also papers in educational sciences that interactive books usually don't increase comprehension of kids (they just play with them instead of depen their understanding). edit: sorry didn't want to sound too critical. The work is awesome (upvote), was just thinking out loud.

tbihl 10y ago
For me this gets to a deeper point. While there are probably some people who innately understand tough mathematical or scientific concepts, it's been my experience that learning is largely about spending enough time struggling diligently with the material until you've built a robust and correct mental model of it. Then on to exercises to see if your model breaks or holds up when strained. If you approach interactive aids and videos and images with this mindset, they can be very helpful. I can also clearly remember times when this was not my approach, and in those times, pictures were just a lot of time saved because I didn't have to read text that otherwise would have been there.
jimhefferon 10y ago
Thank you for the link. I learned from it. I am the author of a widely-used LA text, and have considered adding interactive stuff. But there is a tradeoff. For one thing, it locks you to online, and despite the claims of our IT people, my correspondents (mostly self-learners) do not want online, they want print or PDF (as do I, since the appearance that LaTeX gives me is important to me). For another, the tech has in the recent past changed so fast that maintaining the interactives would be a significant job. I don't mind learning JS to do something good but tying myself to many hours a year responding to bug reports from people on obscure platforms, or using IE6, is not a good use of my life energy. Finally, I had a colleague, a complex analyst, try Visual Complex Analysis and he reported that students did not get it. He is very sober, very caring, very reliable. This starts to make sense of his report.

markbao 10y ago

Amazing resource, but is it still being updated? Last update seems to be from back in July last year.

immersivemath 10y ago
New chapter out today! This time on linear mappings.

eellpp 10y ago

Last year, i saw the videos by 3Blue1Brown and inspired by it went on to read some of the standard text books like linear algebra application by Strang. I had seen the Strang videos earlier but somehow did not follow it through. This time however my perspective was changed. I was approaching the subject through the lens of intuition and simplicity. Wherever i found something challenging, i waited for the next day to again re-do it (because i know that this should not be that complex to understand or the understanding is wrong). And to my surprise, again and again, the difficulty was in my rigidness in understanding. The next day , or even later during the day, when my mind was fresh again, i can reason through the concept and get the intuition behind it. Since then i have seen the Strang videos again and again. Beginning to end. Read the book chapter by chapter and exercises by exercise. And what a delight it had been. And then i jumped upon Joe Blitzstein's probability lectures. What a blast ! Is there a list of teachers like these there, who in the pretext of teaching algebra/probability etc are in reality wiring up our thinking process in ways immaterial to subject they are teaching. Many of us don't want the material to be too casual/layman terms (which hampers self understanding as its no challenging anything within us) and not too rigid (where we cannot break through the challenge).

j2kun 10y ago
> the difficulty was in my rigidness in understanding At what point did you realize this? Like, could you provide a specific example of a topic you thought was hard at first but later came back to and realized was all about the intuition?
- Retra 10y ago
  Not the OP, but... I was just reading Landau & Lifshitz' "Statistical Physics", and can reflect on a series of thoughts that may elaborate on how intuition plays a role in the enjoyment and understanding of complex material. I've been meaning to write it down anyway... On page 3, the book says "A fundamental feature of this [closed system/open subsystem] approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state." When I first read this, I thought "non sequitur, but whatever, I'll continue..." Now, the context of this quote is that the authors are trying to explain why statistical methods work at all. And they said prior that we start with laws that apply to 'microscopic' particles and use statistics to generalize to 'macroscopic' systems. However, the second time I read this, I kept thinking it must be backwards. We didn't understand the motion of (classical) protons before understanding the motion of macroscopic balls. So we had to have been operating under the assumption that the macroscopic laws must apply to microscopic objects, and then require that the must also be reproducible macroscopically through statistical methods. That is, we require that these laws be invariant across the microscopic-to-macroscopic transition. But to do that, we have to use a framework which expresses such a transition. So, for instance, if we are reasoning about the motion of a ball, we have to translate our laws into laws over the motion of some statistical model of the ball. Say, it's center of mass. And with this concept in hand, we could write laws that apply to both the macro and micro worlds, since a 'center of mass' is a macro-micro-scale-invariant abstraction. So we partition the space of all possible laws of nature, and chose to work only in that partition which encodes things we can actually know about nature -- the partition identified by the macro-micro-scale-invariant. So re-reading that passage, it is now not a non-sequitur for me. Now it says "because the interactions with the outside world are so complex, we could not hope to predict their influence. Thus we are justified in using random variables to model their influence, and concepts that derive from the use of random variables to ensure we have complexity-scale invariance when we formulate our laws of physics." And this is not a non-sequitur to me. It follows directly from the meaning of the word "random." Of course statistical methods work when the complexity of a system is indistinguishable from randomness. -- And this whole line of thought generalizes (albeit informally.) For instance, the meaning of words is an invariant across a long thread of contextual translations, and these invariants are used the same way: to partition the space of all possible meanings in such a way that one partition contains all of the 'knowledge' imbued by that word, and thus you can navigate a narrower space of meaning to find the intended and/or correct one. Gives me a certain brand of appreciation for good poetry. Or -- my girlfriend -- who recently told me that she loved algebra but couldn't understand trigonometry. I tried telling her that the algebraic transformations were invariant-preserving operations selected because they conformed to known laws about 'functions' like addition and multiplication which have commutativity and identity laws, and that trigonometry was no different: different functions, but all of the algebraic transformations you needed were selected from the laws of trigonometry with the purpose of maintaining the exact same invariants. (Not that she cared much, to be honest...) And on and on... I could probably talk for days about all the different ways every subject can be reduced to transformations and invariants and how they are used to solve problems.

digitalshankar 10y ago

Can anyone hack this and make a PDF book?

krackers 10y ago
Doesn't that defeat the point of intractability?
- benelot 10y ago
  Interactivity maybe, intractability is not having the property of being tractable. For an algorithm, this means that its performance can not be described in polynomials. But your point is still valid, a pdf does not support animations, thus a pdf would be useless.
  - digitalshankar 10y ago
    I didn't ask PDF for interactivity, i asked PDF for taking notes. The theory is so solid.

ErikBjare 10y ago

I'm a MSc in CS student at the university where the authors (whom I've met a few times) teach. If you have any questions you think I'd know the answer to, ask away!

amelius 10y ago
Perhaps you can let them know their work is being discussed on HN :)
- immersivemath 10y ago
  A good idea! Although now we are already aware. If you have any questions you can ask us here or at immersivemath@gmail.com

techman9 10y ago

I feel as if I'm looking at the future of textbooks. This is absolutely incredible.

darkhorn 10y ago

The order of the words and suffices is like in Turkish. One of my English teachers who has lived in Japan for a time (and she was a Turk) told us that it is very easy to learn Japanese because it follows the same word orders, and vica verse. On the other hand English is very hard becuese there is zero connection.

boramalper 10y ago

Reminded me of Explorable Explanations: http://explorableexplanations.com http://explorableexplanations.com

kowdermeister 10y ago

Jaw dropped. This is exactly what I needed to get some progress with my shader learning :)

corysama 10y ago
You might also like https://thebookofshaders.com https://thebookofshaders.com

proudofyou 10y ago

Maybe I'll try teaching this to my 11 year old sister. Wait, how can I teach calculus to an untalented 11 year old?

francopellicie 10y ago

Al right, where do we get it?

NumberCruncher 10y ago

I wish we had resources like this when I was on the university.

machiaweliczny 10y ago

This looks great! Could some of you share other nice books and video series covering core CS math basics?

blinry 10y ago

If you like this, you might be interested in /r/explorables, a subreddit filled with interactive stuff: https://www.reddit.com/r/explorables/ https://www.reddit.com/r/explorables/

Demcox 10y ago

This will come in handy for my LinAlg exam :3

BooglyWoo 10y ago

Mandelbrot cites his ability to "think in pictures" as fundamental to his process and insights. He offers some interesting reflections and anecdotes on this subject in this interview describing his classes préparatoires aux grandes écoles. Apparently his teacher considered him a total wildcard case who would either flunk the exams or pass with flying colours, because of his habit of approaching everything through geometric intuition rather than symbolic manipulation. https://www.webofstories.com/play/benoit.mandelbrot/6 https://www.webofstories.com/play/benoit.mandelbrot/6

mathgenius 10y ago

To me this book looks like a whole bunch of equations, with some fancy graphics sprinkled on top. And, far too many equations! Linear algebra is much more elegant (simple) than this. To pick one example, they define the inner product using the cosine of the "smallest angle between the two vectors." Sure if you want to calculate a number (an inner product in two dimensions), and you happened to know the angle in question, this might be helpful. But otherwise it completely obscures everything else about the inner product. How does an interactive graphic help you understand wtf is a cosine doing in this equation? What is a cosine anyway? Where is the graphic for that? This just seems too backward and over-done to me. But go ahead and test it on some newbies, maybe I'm totally wrong here.

jacobolus 10y ago
More importantly, the definition of the cosine is given by projecting a unit vector onto a given directed line, and then measuring the length of the projection. The way to find a given cosine of a given angle between two arbitrary vectors is by taking the dot product of the two vectors and then normalizing by their lengths. Using the cosine to define the dot product is precisely backwards.

sp4ke 10y ago

You might be interested by this curated list of explorable explanations [1]. It has a good section on math. [1] https://github.com/sp4ke/awesome-explorables https://github.com/sp4ke/awesome-explorables

jarek83 10y ago

Great content, makes it much easier to get through concepts. But as most math learning resources, it lacks examples of their practical application. "The law of cosines is a very useful formula to know" - fine but why? Most learners can't imagine where they can apply what their teachers say, which brings it down to learn useless-for-them terms, needed just to pass exams. Some basic examples like these would make it much more reasonable to learn: http://study.com/academy/lesson/solving-real-world-problems-using-the-law-of-cosines.html http://study.com/academy/lesson/solving-real-world-problems-...

msaharia 10y ago

Q: How would one go about creating such animations? What's the best workflow? Found the figures were slow to load. Great resource, though!