
Immersive Linear Algebra – textbook with fully interactive figures (2015) - sebg
http://immersivemath.com/ila/index.html
======
eellpp
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
> 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
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.

~~~
noobiemcfoob
> 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.

I find myself partial to this type of world view too. I believe it is part of
the appeal of functional programming, at the basest level, to shape the
programming model into transformations (functions) and invariants (state).

------
laretluval
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=PLZHQObOWTQDPD3MizzM2x...](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)

~~~
teperpencoli
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
Wasn't he hired by Khan Academy recently? It's great if that helps a wider
audience to benefit from his work.

~~~
ludicast
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.

------
ktta
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!

~~~
swiley
Well written software doesn't seem to go stale, or at least not quickly and
poorly written books do.

~~~
ktta
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

------
Dangeranger
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
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.

------
yellow_postit
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
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
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
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-a...](http://www.hnwatcher.com/r/984396/Embed-interactive-Python-
anywhere-on-the-web)

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
Thanks for these great references! I will definitely have a look at them and
see if I can use them for my purposes.

------
blt
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.

------
dang
Discussed in 2015:
[https://news.ycombinator.com/item?id=10183725](https://news.ycombinator.com/item?id=10183725).

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

------
kgarten
on a tangent, I was just reading [https://medium.com/@dominikus/the-end-of-
interactive-visuali...](https://medium.com/@dominikus/the-end-of-interactive-
visualizations-52c585dcafcb)

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.

~~~
jimhefferon
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.

~~~
Myrmornis
I'd be interested to know your thoughts on the Essence of Linear Algebra
videos by Grant Sanderson. What role do you see material like that playing?
(They are fantastic if you're not familiar)
[http://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xV...](http://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)

~~~
jimhefferon
Yes, I admire them. I have some technical quibbles, as a fellow educator, but
I admire them a lot.

------
mathgenius
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
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.

------
ydmitry
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)

------
mkl
Does anyone know more about how the figures were developed?

I found
[https://www.lth.se/fileadmin/lth/genombrottet/LUkonf2015/41_...](https://www.lth.se/fileadmin/lth/genombrottet/LUkonf2015/41_Stro__m_etal.pdf)
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
Is there anything in particular that you would like to know?

------
jarek83
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-...](http://study.com/academy/lesson/solving-real-world-problems-
using-the-law-of-cosines.html)

------
sp4ke
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)

------
blinry
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/)

------
techman9
I feel as if I'm looking at the future of textbooks. This is absolutely
incredible.

------
BooglyWoo
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)

------
ErikBjare
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
Perhaps you can let them know their work is being discussed on HN :)

~~~
immersivemath
A good idea! Although now we are already aware. If you have any questions you
can ask us here or at immersivemath@gmail.com

------
darkhorn
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.

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

~~~
immersivemath
New chapter out today! This time on linear mappings.

------
mrkgnao
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...](https://terrytao.wordpress.com/career-
advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/)

~~~
Synaesthesia
You have to show her the videos from 3Blue1Brown. They're so good

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

------
boramalper
Reminded me of Explorable Explanations:
[http://explorableexplanations.com](http://explorableexplanations.com)

------
msaharia
Q: How would one go about creating such animations? What's the best workflow?
Found the figures were slow to load.

Great resource, though!

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

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

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

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

~~~
corysama
You might also like
[https://thebookofshaders.com](https://thebookofshaders.com)

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

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

------
Demcox
This will come in handy for my LinAlg exam :3

------
digitalshankar
Can anyone hack this and make a PDF book?

~~~
krackers
Doesn't that defeat the point of intractability?

~~~
benelot
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
I didn't ask PDF for interactivity, i asked PDF for taking notes. The theory
is so solid.

------
aligajani
Wow, this is great.

~~~
francopellicie
Al right, where do we get it?

