
An Intuitive Guide to Linear Algebra - nkurz
http://betterexplained.com/articles/linear-algebra-guide/
======
enahs-sf
It seems like college Linear Algebra expeiences fall into one of two buckets:
either you took a "Linear Algebra / Differential Equations" class where you
learned no theory and saw a bunch of practical examples and did problem sets
that were routinely north of 10 pages, OR you took a highly theoretical course
where everything was far too abstract for your first year of college level of
comprehension.

I took the first route and retained none of the information, despite doing
well in the course. Then I took computer graphics and cried all the way home.

This link seems like a happy middle ground.

~~~
nextos
I found Linear Algebra Done Right by Axler to be a fantastic theoretical text
that makes everything click together from day 1. It's derived from his paper
Down with Determinants.

He starts by explaining that linear algebra is about studying the properties
of linear transformations over vector spaces, and then goes on to explain what
this means. In many theoretical books it takes ages to get this message, and
in some practical ones you get lost in a sea of matrices without even
understanding what a matrix actually is. Furthermore all his proofs are very
slick.

~~~
tptacek
Anyone want to study group this book? I did Strang a few years back, along
with the OCW lectures, but it wasn't very proofs-centric.

~~~
pakled_engineer
MIT has some more proof centric courses
[http://ocw.mit.edu/courses/mathematics/18-700-linear-
algebra...](http://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-
fall-2013/)

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skierscott
Linear algebra is cool, no doubt. This blog post is certainly well written and
explains how to think of matrices as functions.

I have written a blog post[1] that describes matrices as functions. This blog
post is titled "Correcting common mathematical misconceptions" and walks
through N dimensions, linear functions and linear algebra. After explaining
what linear functions are, I mentioned nonlinear functions and how they often
don't have a closed form solution and why mathematicians find bounds.

[1]:[http://scottsievert.github.io/blog/2014/07/31/common-
mathema...](http://scottsievert.github.io/blog/2014/07/31/common-mathematical-
misconceptions/)

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eli_gottlieb
>We’re getting organized: inputs in vertical columns, operations in horizontal
rows.

OH! Bloody hell, so _that 's_ why we write it like that!

>The eigenvector and eigenvalue represent the “axes” of the transformation.

>Consider spinning a globe: every location faces a new direction, except the
poles.

>An “eigenvector” is an input that doesn’t change direction when it’s run
through the matrix (it points “along the axis”). And although the direction
doesn’t change, the size might. The eigenvalue is the amount the eigenvector
is scaled up or down when going through the matrix.

Very nice! I'd had an algebraic (ahaha) understanding of eigenvectors/values
before, but hadn't a geometric intuition. Thank you. Now I can neatly imagine
why the eigenvector is orthogonal/perpendicular to the "direction" of the
transformation.

Determinants, too.

Excellent article.

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quickquicker
I've always been partial to this: [http://blog.wolfire.com/2009/07/linear-
algebra-for-game-deve...](http://blog.wolfire.com/2009/07/linear-algebra-for-
game-developers-part-1/)

~~~
iamcreasy
They stopped posting these kinds of stuff. Sad.

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IndianAstronaut
A lot of programming puzzles companies hand out can be solved with linear
algebra. One company I interviewed at gave me a pretty complex problem, solved
it with linalg. Didn't get the jib since I didn't use the usual data
structures, but it was still a fun experience.

~~~
ChristianGeek
Your use of linear algebra over what was expected would have significantly
increased your chance of getting an offer if I'd been interviewing.

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aswanson
I wish someone would better explain Lie algebra.

~~~
throwaway5752
I'll would request the same for p-adic numbers, while they're at it!

~~~
Chinjut
p-adic numbers are actually quite simple; so simple, I've actually seen
children invent it on their own!! They're just like ordinary decimal notation
numbers, except instead of allowing these to extend infinitely to the right
(as in 3.14159...), you allow them to extend infinitely to the left. So, in
addition to familiar numbers like 45 or 67.8, you also have numbers like
...95141.3 or ...999 or what have you.

You add and multiply these according to the same rules you're already familiar
with. So, for example, if you add 1 to ...999, the last digit of the output is
0, and you get a carry. Making the second to last digit 0, with another carry.
And so on and so on, making the result ...0000 over all. Thus, ...999 acts
like -1.

(If we were working in base two, this last example would be just like the
"two's complement" you are perhaps familiar with from computer arithmetic!)

In fact, most discussion of p-adic numbers isn't done in base ten. Instead,
people typically focus on p-adics in a prime base (hence the p). Why? Because
in a prime base, you will find that every nonzero p-adic number has a
multiplicative inverse, which is very convenient (while in a composite base,
you will find that sometimes nonzero numbers multiply to zero). But there's
nothing actually stopping you from making use of the notion for non-prime
base, should you be interested in doing so; the notion is still perfectly
coherent. (That having been said, another reason mathematicians focus on
p-adics in prime bases is that the Chinese Remainder Theorem essentially
allows one to reduce the study of all other bases to this case.)

There's a lot of beautiful further theory to explore here, but again, the
basic idea is hopefully quite simple. Let me know if you have any questions
and I'll be happy to try explaining further or more clearly.

~~~
zwegner
Interesting. Very similar to the way ints are represented in Python (since
they have arbitrary length, negative numbers are represented with infinitely
many 1's in the most significant bits).

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allending
If you like this, then go read the No bullshit guide to math & physics after
this: [http://minireference.com](http://minireference.com).

Love the trend of indie publishing that has been exploding in the past 2 to 3
years. The biggest issue is that the marketing & design is often mediocre.
Content wise, completely awesome though. I lost interest in books from larger
publishers years ago.

~~~
wnevets
Is this book worth getting for someone who doesnt remember anything about math
from high school and beyond?

~~~
allending
I think so – it covers the fundamentals. Numbers, Algebra, Functions. Not
exactly in depth, but if you've done high school math you should quickly
remember and be able to get to the fun stuff.

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agumonkey
I always loved the fact that matrices encode operation on matrices (see
comment#1 [http://betterexplained.com/articles/linear-algebra-
guide/#co...](http://betterexplained.com/articles/linear-algebra-
guide/#comment-124145) ). Very meta-circular, Ycombinatorish. Would that mean
Linear Algebra is homoiconic ?

~~~
reikonomusha
Having group or ring structure isn't too surprising and isn't too intimately
related to the idea of metacircularity or the Y-combinator. It's actually more
interesting to really see that matrices encode linear transformations, and
multiplication is function composition.

~~~
agumonkey
Yes, I can't recall where I first read this (someone's comment on HN most
probably) but it was very enlightening.

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niels_olson
Perhaps I don't recall my linear algebra well (we called it engine math), but
there's a hell of a lot more to it than just passing inputs through a series
of operations. Linear algebra is more about the methods used to design very
elegant models and transforms and understanding the limits of those models and
transforms.

~~~
Ar-Curunir
No, those are the _applications_ of linear algebra.

Linear algebra itself is just a study of vector spaces and linear operators
and transforms and such.

~~~
niels_olson
For context, I don't think I ever knew the thing we were studying was called
linear algebra, so it took me another 17 years to realize that is the common
name for what we studied in that class. Now I'm wondering if maybe one of the
intents of the small engineering-focused school's tightly integrated math-
science-engineering curriculum was to avoid an introductory linear algebra
class. We had to understand matrices and vector spaces earlier for other
classes and didn't do what you call the "applications" of linear algebra until
sophomore year.

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interdrift
I took linear algebra 2 semesters ago.It was awesome! Highly recommend
learning it, I use it on a daily basis in my work in 2D and 3D!

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dj-wonk
I find it non-intuitive that many of the input-output diagrams flow from right
to left.

(We all know that causality only flows left-to-right.)

~~~
SAI_Peregrinus
That's because matrix multiplication is noncommutative, and it's traditional
in linear algebra to write the inputs matrix on the right of the operations
matrix. Just like in normal algebra, you write 5x+7y instead of x5+y7. So the
picture matches up with the notation.

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wodenokoto
Has anyone bought their calculus course? is it worth it?

