
Immersive Linear Algebra (2016) - reverse
http://immersivemath.com/ila/index.html
======
nimonian
Without having looked at the text in depth, I applaud the idea here - I'm
convinced the web browser can help us improve upon maths textbooks printed on
paper, but there aren't many 'hypertextbooks' like this in existence (partly,
I think, because most people positioned to author a maths textbook want to use
latex; and most people who are able to develop a web app aren't positioned to
author a maths textbook).

I'm interested to know how the authors funded this, and if they have any data
about the impact it has had.

~~~
jimhefferon
For my Linear Algebra
([http://joshua.smcvt.edu/linearalgebra/](http://joshua.smcvt.edu/linearalgebra/)),
the thing that has held me back has been that the tech changes every few
years. I can't keep revising something that I give away for free. In contrast,
the LaTeX in which I wrote the book is 98% unchanged since the early 90's.

Perhaps the tech is settling down, I'm not sure.

~~~
nimonian
Perhaps this is a reason why there are some nicely designed apps with not-so-
great content, and lots of books with legendary content: the whole ecosystem
around app development isn't very friendly to the long-term process of writing
a classic text.

It's also hard to focus on design, coding and content solo, and writing great,
cohesive content tends to be a private endeavour. I'm not surprised to see
immersivemath.com is written by a team.

I know (through fond memory!) your texts have some interactivity in
hyperlinking to solutions and I found that very useful, but making it work
must have taken time and been a bit of a headache. Extrapolate that to writing
a webapp and it's easy to imagine you would never finish.

And yes, the words 'legendary' and 'classic' are direct compliments to you:
your Lin Alg sits next to Spivak's Calculus on my bookshelf. They are both
immortal.

~~~
jimhefferon
Thank you, I appreciate it.

------
jahbrewski
For another great source of open source textbooks I highly recommend OpenStax
([https://openstax.org/](https://openstax.org/)). I am currently re-learning
calculus and their Calculus text is great. As an aside, I'm currently working
through mathematics I wish I had taken in college (I was a biology major),
starting with Calculus, and would love to connect with anyone else on a
similar mathematical journey!

And for even more open source textbooks:
[https://open.umn.edu/opentextbooks/](https://open.umn.edu/opentextbooks/)

~~~
scarecrowbob
I've been doing Khan academy most days (like, 28 of 30) for the last year.
It's been super friendly and I've moved through the algebra, geometry, trig,
precalc, and differential material... I'm currently working through some
integral calc stuff. I'm planning on doing the stats and linear parts of kahn
before trying to take some of the online college-level courses; Sal Kahn is
super great as a teacher but I am starting to feel like I can now benefit from
some more difficult to digest material.

My hope is to generally increase my math abilities so I can do some
electronics design classes and some study about how to start applying ML/ AI
tools.

But I've found studying math to be quite fun and useful in its own right. It's
fun and I feel like it's helped my general intellectual abilities.

~~~
dexter_t
I too would like to work my way up to calculus and beyond.

What sequence of courses do I take on the site to get to calculus, starting
from the beginning with the basics?

~~~
pault
I had a humbling experience last year when I browsed the available courses
until I found one that only contained math I'm already familiar with and I had
to go all the way back to fourth year elementary school. :/

~~~
dexter_t
How were you able to determine what level you were at?

~~~
pault
I started at precalculus and worked my way back until I found a video that
didn't assume I already knew something I don't already know.

~~~
ambicapter
Could be that you know a lot of maths at a higher level than that, but your
education just has a lot of holes in it.

~~~
pault
If only that were true. :) I only know cross multiplication and basic
arithmetic, excepting long division.

~~~
jahbrewski
We all start somewhere! One of the beauties of math is it’s hierarchical
structure. Once you figure out where you’re at, it’s just a matter of
progressing further up the hierarchy of abstraction (spoken by someone who has
only made it to Calculus in the hierarchy)

~~~
voltagex_
I really want to make a site that expresses this like a skill tree you'd find
in an RPG or in Civilisation.

~~~
nimonian
Duolingo has something like this. I rather like it.

------
b_b
This seems like one of the best uses of computers for educational media, one
of the ways digital instruction can really shine through compared to physical
text which can often require supplemental videos or help from a teacher to
elucidate a concept.

Feels very reminiscent to me of 3B1B videos [1] on topics that use visuals to
elucidate math topics more clearly as well.

[1] =
[https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw)

~~~
aidos
3Blue1Brown's Essence of linear algebra series is the go to place for learning
this material. I've included the link to the series below so people don't get
lost in all his other wonderful videos along the way.

[https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...](https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)

~~~
throway88989898
No offense to 3b1b, the production quality is impressive and the content is
effectively insightful.

However, it is ultimately a very limited linear format covering the topic only
lightly.

~~~
tokyodude
I see you used a throwaway because there are so many fans. I might not be as
harsh but while I enjoyed the pretty animations they didn't actually help me
learn linear algebra. They're still full of jargon written for people who
already know the topic.

------
throwawaymath
Upvoting this because I think it's really important that we start getting more
open source textbooks which specialize in exposition[1]. That being said, I'd
make a different arrangement of the material.

Here are a few thoughts:

1\. This book embraces the modern take on a rigorous course in linear algebra,
which I disagree with. That is to say, it covers vector spaces front and
center in the first chapter. Conversely, systems of linear equations (Gaussian
elimination) and corresponding notions like row reduction of
coefficient/augmented matrices are pushed back several chapters. You see this
in popular "flagship" textbooks like Axler's _Linear Algebra Done Right_
because they're prioritizing the underlying theory. I think Gaussian
elimination should be covered first because it motivates the material - you're
never going to cover Linear Algebra in a way that makes all the pieces fall
together in a purely straightforward dependency.

But if you first cover systems of linear equations, you actually motivate the
purpose of vector spaces. Moreover, many (most?) proof-based problems in
linear algebra reduce to solving a system of linear equations (and therefore
reducing a matrix) which is modeled after the thing you're trying to prove.
What's the basis of a null space of this linear transformation? You can solve
it with row reduction. Is this spanning set a basis of this vector space?
Solvable through row reduction. You get a lot of equipment for solving vector
space problems if you build up the row reduction scaffolding to a minimal
level before vector space coverage. This does not have to temper the
theoretical rigor of later, abstract material; if anything it grounds and
augments it.

This is why I'm a huge fan of the way that Hoffman & Kunze cover this
material. That is a _heavily_ theoretical book, which easily surpasses
undergrad-level linear algebra in later chapters. A couple of decades ago it
had the spotlight as the really good linear algebra book (it displaced Halmos'
_Finite Dimensional Vector Spaces_ , and was itself displaced by Friedberg or
Axler). But it motivates a very rigorous coverage of vector spaces by starting
with Gaussian elimination, matrix representation of systems of linear
equations and (reduced) row echelon forms. If you don't cover this first, you
have to either assume the student knows it already or you resign yourself to
wading through several chapters of potentially unmotivated abstraction.

2\. In my opinion, linear maps should be covered directly after vector spaces.
They're a logical succession of the same concept - after you've taught linear
combinations, linear (in)dependence, spanning sets, bases, subspaces, and
related theorems, it's very easy to jump right into functions _between_ vector
spaces. It's just an expansion on all of these things. I would therefore
rather see chapter 9 immediately following chapter 2. And much like my prior
point, linear maps are an abstraction that empowers a lot of the other ideas;
this is mainly because so many things are representable as a linear
transformation (much like they're representable as a matrix).

3\. The rank is a very important concept, but I would argue it should be
subsumed into the chapter on matrices. You can cover anything that doesn't fit
cleanly in the matrices chapter in the chapter on linear maps, since you'll
need to do that anyway before you cover the dimension theorem. That might be a
good argument for simply covering matrix multiplication and addition in
prerequisite material and subsuming all of matrices under Gaussian elimination
and/or linear maps too...

4\. I'm a big fan of the prerequisite material covering basic trigonometry.
Trigonometric functions are a good source of exercises for vector spaces. I
would like to see an expansion which does minimal coverage of naive set theory
(sets, subsets, unions/intersections, etc), functions (injection, surjection,
bijection) and polynomials. If you cover the basic definition and arithmetic
of polynomials here, you save yourself some space when you cover Lagrange
interpolation and factorization later on. It's probably a lot to ask for, but
some coverage of differentiation of polynomials would also be useful because
the theory of vector spaces gets a lot richer when you incorporate exercises
which use a bit of calculus/analysis. In particular, using vector spaces of
functions and linear maps of differentiation give you a lot of leeway to show
how abstract vector spaces can be without going nuts on the moon math.

5\. I've received pushback on this before, but I think it's really important
to cover fields as their own subject before covering vector spaces. If you're
going to cover vector spaces rigorously, you really need to cover the field
axioms. Scalar multiplication doesn't make sense without first defining a
field (you can handwave it, but it's easy to make subtle mistakes - like
missing that the set of all integer $n$-tuples _cannot_ comprise a vector
space). Likewise you ordinarily need your linear map to agree on a field to be
well-defined. This coverage would ideally also establish the basic facts of
the real and complex fields, which are almost entirely the fields you work
with in undergrad linear algebra.

6\. I don't see any exercises. A good math textbook challenges its reader to
internalize each chapter's material by substantially using it to prove further
material. By not including exercises, this text becomes more of an illustrated
monograph. That's totally fine if it's what they're going for, but I think
there's a huge opportunity to tailor it even further with targeted exercises
that expand on the theorems in each chapter.

Note that I'm mostly nitpicking here, even if I think my critiques have merit.
The exposition of material is much more important than the arrangement of
material, as long as things are building on each other rigorously. I'm not
trying to say the textbook is bad; this is just a stream of consciousness
about my opinions on arranging linear algebra for maximal pedagogy. You
generally have to supplement textbooks in order to get a rich understanding of
the subject matter anyway.

On the plus side, I'm a huge fan of the way they present vectors
geometrically. That's a huge win for illustration/animation purposes. I do
think there's a missed opportunity to make a sexy illustration of transforming
a system of linear equations into coefficient, unknown and constant matrices;
then from there illustrating the transformation of the augmented matrix into
its reduced row echelon form.

__________________________________

1\. For pretty good open source, undergrad-level textbooks in Abstract and
Linear Algebra respectively, see
[http://abstract.ups.edu/aata/](http://abstract.ups.edu/aata/) and
[http://linear.pugetsound.edu/html/fcla.html](http://linear.pugetsound.edu/html/fcla.html)
respectively.

~~~
crdrost
I think the problem is a tension between what’s valuable to us in learning and
what's valuable to us after learning.

So when I was tutoring, I noticed that my “standard approach” failed: a kid
would have a homework problem, I would say, “Here's how you want to think
about that problem, it’s (say) a conservation of energy problem, calculate the
energy before and after...” and it fostered no real learning! I thought the
problem was that I was subsidizing laziness—stepped back a bit but gave
lessons after some struggling—and that didn’t hurt much but it didn’t help
much either. They were motivated, just the didactic structure was not working
for them. I started to ask questions, and that was a bit more helpful, but
found that I myself was somewhat unguided about how to accomplish “Socratic
teaching” so that they could benefit from it—if you can believe it, my default
approach was to try to introduce the same theory ideas as questions from the
start, “would conservation of energy apply here?”—so no wonder I only got
mixed results: it was basically the same thing.

But questions gave me a way to get off that map, because I started to ask
tutees to give me _expectations_ and _examples_ , and somehow those worked
well. “What can we expect that this system does, within a few orders of
magnitude? Can you give me some examples of similar systems and how they
work?”

I think that in order to learn abstract things, we need to first have a bunch
of examples in our heads, things that all kind of fit together but do not make
sense. The abstraction unifies the examples, helps us remember them and fit
them together. We can use each example to test the abstraction for fit, to
massage our cognitive ability from one context into another.

When we build a building, we need frameworks and scaffolds to build it. When
the building is freestanding, all of that effort of building the scaffolds can
be undone, the scaffolding comes down, it's no longer necessary. We go back to
a textbook and we want a picture-book: show me the building, and just the
building, and nothing but the building. Let me feel lushly indulged by its
architecture, by photographs that go into detail in the really rich areas—I
want indulgent artistic experience, because that's what the building is!

We have a constant tension in textbooks between the how-to-manual—spartan,
practical, containing all of these scaffolding steps and nitty-gritty–and the
coffeetable book—rich, whimsical, pop-simplified.

Griffiths’ _Introduction to Quantum Mechanics_ begins with a prologue to this
effect, though I thought it was just a quantum quirk at the time I think it's
actually much more general now. He says [these will not be exact quotes] that,
“This book will teach you to _do_ quantum mechanics because I strongly believe
that this has to come before any sort of strange discussions about what
quantum mechanics _is_.” It’s one step between Feynman’s “don’t even try to
understand quantum mechanics, just shut up and calculate” and Searle’s
response “I’m sorry, I am a philosopher and literally the only thing anyone
pays me for is to try and understand things like quantum mechanics.”

And I think that is where I want to focus my teaching efforts, that every
textbook should kind of have the appendix coffee-table section, I’d even like
to carve out a new name for it and call it an “abridgement” or so, at the end.
The book _Mazes for Programmers_ has an appendix like this of “let me
summarize all of these maze generation algorithms and their essential
properties and their essential approaches” that I _really_ loved, it makes it
an absolute joy to keep this thing on the bookshelf and come back to it time
and time again. The main text has the “Here’s how to _do_ it” information, the
appendix has the polite overview of what was just built that is great for
returning to.

~~~
mncharity
Incoming tech seems about to change the texts-and-tutoring constraint and
opportunity space. I wonder if it's time to start thinking and exploring
ahead?

Consider an art project, where you sit in a booth, and watch an interesting
video drama. But why is it interesting? The video is a graph of segments, like
an old "choose your own adventure" story. And there's an eye tracker watching
you. So if you're interested in characters A and B, the story mutates to
emphasize them.

Consider a one-on-one tutor on a good day. Noticing a student engaged and
enthused, they might reorder content on the fly to leverage that. Might
emphasize different aspects, and alter presentation, based on observed
interests and weaknesses. Or consider working with a young child, watching
them read word by word, noticing their where they hesitate and frown, probing
for their thoughts, adjusting difficulty, providing a path of development.

What if that was math content rather than a drama or children's book? What if
we could do this at scale? Eye tracking is just one tech coming in on the
coattails of VR/AR. A setting for personalization AI is another.

What if saying "the best way to organize and present this topic in a
textbook", becomes like saying "the Capital mandates, that every teacher of
this grade, will today all teach the following lesson, by saying the following
words, regardless of local context"? While not on a national level, that is a
real thing.

What might it take to start encoding an adventure graph for linear algebra?
The "oh, if you like this perspective on this topic, you might like this
similar perspective on this other topic"?

Or if we don't have the tooling for that yet, can we start thinking about the
tooling? Or fruitfully do something else now, in preparation for opportunity?
Perhaps Kahn academy problems in more flavors, in a richer graph? ML-based
textbook aggregation, synthesis and retheming? Perhaps it's all not ripe yet.
But something else, maybe?

~~~
edflsafoiewq
I feel a bit bad for saying this, but I don't think the interactive
visualizations here really contribute very much. Yes, you can move the
vectors, but the point is already made by the static picture.

Similarly, you can already traverse, not only a single math book in a non-
linear order, but any number of different books and other sources
concurrently, and this is how everyone I know of already learns. Many
textbooks already have a dependency graph in the beginning showing how you can
read the chapters! So every person is already traversing their own
personalized "adventure graph" for linear algebra and will be throughout their
entire education. It is rather the idea of a totalizing tech solution that
will be perfect for everyone that smacks of central planning.

~~~
mncharity
Hmm, I hope the "centralized planning" story wasn't distasteful. I was
thinking of the stark contrast between say my writing a learning progression
for category theory, versus say pointing out to a toddler that their
observation about a game piece on a path, generalizes to any finite loop,
including time of day, or a simple parking lot.

So let's see, possible contributions from interactive visualization to
teaching linear algebra? Very not my field. And it's been decades for me. And
my exposure to math education research is limited. So I don't recall what
challenges, misconceptions, and failure modes are faced there. So, all I can
offer is a handwave: perhaps a hands-on version of some 3Blue1Brown video?

Apropos "this is how everyone I know of already learns", at least for science
education, this describes very very few K-13 students. Even among freshmen at
a first-tier university. I'd be surprised if math was significantly different.
Surprised but very interested.

Apropos "Many textbooks already have", yes... progress is often not something
startlingly novel, but doing something we've already recognized as desirable,
but doing it faster, better, more thoroughly, more cheaply, more consistently,
for more people, etc.

Perhaps it might be more useful to think of tutoring others, rather than
learning oneself? Dropping on someone a pile of texts, and telling them "find
the corresponding sections yourself, work past the differences of notation,
when you you think you might have a misconception, try googling the math
education research primary literature to find how to deal with it, ...", well,
hmm. What are the learning experiences we would ideally wish for each student,
and can we use incoming tech to deploy less ghastly approximations of that.

------
tabtab
I thought matrices were pretty cool when I took Linear Algebra in college.
However, I hated doing the computations by hand; it makes one really
appreciate computers.

------
photon_lines
I recently released my own review notes which try to provide a brief overview
of some of the main linear algebra topics in an intuitive / geometric manner
in case anyone is interested: [https://github.com/photonlines/Intuitive-
Overview-of-Linear-...](https://github.com/photonlines/Intuitive-Overview-of-
Linear-Algebra-Fundamentals)

------
saivan
I've been working on this for the last two years, I've just started writing
content for it :)

[http://treena.org](http://treena.org)

~~~
outime
The website looks a but weird on mobile [0]. It’s still readable but probably
removing the left image would help already.

[0] [https://imgur.com/dzidyuV](https://imgur.com/dzidyuV)

------
Aissen
This reminds me of the mobile game Euclidea, which is a puzzle game where one
must build geometric figures with as few operations/tools as possible:
[https://www.euclidea.xyz/](https://www.euclidea.xyz/)

It's really well designed, give it a try !

------
iamcreasy
There is a similar interactive linear algebra book from Georgia tech :
[https://textbooks.math.gatech.edu/ila/](https://textbooks.math.gatech.edu/ila/)

Highly recommended.

------
ghosthamlet
Another interactive math textbook:
[https://mathigon.org/](https://mathigon.org/)

~~~
nimonian
Wow. That is an amazing piece of work.

------
MentatOnMelange
This is fantastic! I'm planning on taking linear algebra in the fall and a
glance over this makes me think it will save me tons of time searching youtube
and etc for explanations of trickier concepts.

~~~
generaljellyj
3Blue1Brown's youtube channel is great too for visuals and theory. I taught
myself linear algebra pretty easily combined with this book.

------
notthemessiah
Reminds me of [http://setosa.io/ev/eigenvectors-and-
eigenvalues/](http://setosa.io/ev/eigenvectors-and-eigenvalues/)

------
pmarreck
20 years ago, when web browsing was still newish, I would have expected
HUNDREDS of interactive-learning sites like this to have been set up by now. I
am very glad to see this!

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justplay
Thank you so much to ease my life.

Do anyone know any such type interactive books which is related to ML, i don't
mind purchasing.

------
giovanni_m1
Thanks for the link. I have been meaning to brush up my Linear Algebra for
some time now, I’ll give this a serious try.

Side note: anyone else notice that if you scroll up or down really fast (in
mobile) the spinning logo starts to go faster & faster. Cool Easter egg!

------
cwbrandsma
I really like the concept...but where do people pronounce algebra that way? It
sounds like a computer trying to sound normal (to my ear. I'm more of a Al-Ja-
bra kind of guy, not Al-Ga-bra)

------
slowhadoken
I need a resource like this for group theory.

~~~
throwawaymath
Try [http://abstract.ups.edu/aata/](http://abstract.ups.edu/aata/). I have no
affiliation with the authors, but I think it's really quite good. Symmetries
in particular have a lot of illustration.

~~~
slowhadoken
Thank you! I'll check it out.

------
haskellandchill
I want interactive proofs to learn linear algebra. I'm trying to build
something, it's daunting.

~~~
mhh__
Interactive proofs?

~~~
haskellandchill
Yes. Unfortunately that term is taken by some cryptography thing. A proof
assistant or interactive theorem prover to be more precise.

------
pylus
Great idea.

Learners can interact with book.

------
michaelsbradley
Is there anything similar available for Tensors?

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sexy_seedbox
Ouch, their Forum is full of spam.

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FrankDixon
yes! interactivity is where it is!

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rodiger
Is this still being worked on?

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poojapanwar27
test

