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Graphical Linear Algebra

graphicallinearalgebra.net

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Re: Graphical Linear Algebra

#2

Post by siraben »

When I took a course on linear algebra, the professor would introduce the notion of vector spaces and linear transformations first before talking about matrices, and actually one can derive the matrix multiplication algorithm from the specification of how it must correspond with composition of linear transformations. I thought that the algebraic notation is very concise and appropriate for linear algebra.

While new notation like this is interesting, I do wonder how useful it is from a pedagogical standpoint for linear algebra. Category theory OTOH is one of the areas of mathematics with diagrams rule over equations because a single diagram can encode so many equations.

For excellent introductions to linear algebra I recommend Linear Algebra Done Right[0], which Springer offered for free for a while (it doesn't seem to be anymore though, contact me if you want the PDF).

[0] http://linear.axler.net/

Re: Graphical Linear Algebra

#4

Post by CharlesMerriam2 »

Meh. Sorry.

Start with [Essence of Linear Algebra](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...) by [3 Blue 1 Brown](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw) for an intuitive overview of all the major concepts.

While working through a full Linear Algebra course, it is important to develop your own software to do things. That is, some sort of command tool to enter matrices and run operations on them. As you go through a course, you will keep adding operations.

You should work on some of the most common tricks, like using 4 by 4 matrices for computer graphics and understanding what a PCA does.

Re: Graphical Linear Algebra

#5

Post by spekcular »

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in.

This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something interesting but end up just rehashing standard mathematics in opaque ways.

I really wish this would stop. Fortunately (unfortunately?) I've been down this road a few times and know to avoid getting sucked in, but I can easily see a bright, curious person wasting a lot of time before realizing that the content is merely linguistic, not mathematical.

Is (for example) category theory a useful organizational tool in certain abstract branches of mathematics? Sure (those branches being, basically, algebraic topology and algebraic geometry). Is it a grand unified theory of math? No, it's just some useful vocabulary. Most mathematicians go their entire lives without writing a paper mentioning categories. Getting excited about category theory is like getting excited about matrix notation – useful, sure, but not where the meat is.

I also find any claims that category theory is relevant to the average working programmer to be dubious at best.

(And yes, I realize the graphical notation presented in this article is not category theory. I am making a broader point about certain kinds of articles I see, which also applies here.)

Re: Graphical Linear Algebra

#6

Post by guerrilla »

enriquto wrote: post #3

It would be nice to have a single-page complete summary of this notation. The whole text is way too verbose and boring (at least the first two chapters).

There's a paper linked to here which he says contains all the "spoilers": http://arxiv.org/abs/1403.7048

> If you are impatient and want to get to the conclusion quickly, here’s the link to our paper where the basic theory is explained (Spoiler warning!). We will also discuss several applications of graphical linear algebra that have been developed in recent years.

Re: Graphical Linear Algebra

#8

Post by edflsafoiewq »

The usual notation for a linear transformation

  X' = aX + bY
  Y' = cX + dY
minimizes irrelevant information. But the diagram for one seems to bring it to the fore: a diagram is basically a system of fully parenthesized, unsimplified expressions eg

  X' = (3X + (2(X+4Y) - 6X)) + Y
  ...
In fact, it contains even more information than that, since it also tells you if, in (X+Y)+(X+Y), you are supposed to compute (X+Y) once and reuse the value or compute it twice.

Re: Graphical Linear Algebra

#9

Post by s1t5 »

CharlesMerriam2 wrote: post #4

Meh. Sorry. Start with [Essence of Linear Algebra]( https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... ) by [3 Blue 1 Brown]( https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw ) for an intuitive overview of all the major concepts. While working through a full Linear Algebra course, it is important to develop your own software to do things. That is, some sort of command tool to enter matrices and r…

As much as I like 3blue1brown, watching those videos is not a good way to learn. It's maybe a good way to get some intuition about a topic and much closer to the "infotainment" category. A good textbook + solving problems + maybe writing code works much better for actual learning.
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