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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#51

Post by tenacious_tuna »

tptacek wrote: post #29
andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just…

> I didn't think any part of linear algebra was boring.

My formal linear algebra course was boring as hell, to me. The ~4 lectures my security prof dedicated to explaining just enough to do some RSA was absolutely incredible. I would pay lots of money for a hands-on what-linalg-is-useful-for course with practical examples like that.

Re: The Little Book of Linear Algebra

#52

Post by tptacek »

tenacious_tuna wrote: post #51
tptacek wrote: post #29

Earlier quoted context omitted.

I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just…

> I didn't think any part of linear algebra was boring. My formal linear algebra course was boring as hell, to me. The ~4 lectures my security prof dedicated to explaining just enough to do some RSA was absolutely incredible. I would pay lots of money for a hands-on what-linalg-is-useful-for course with practical examples like that.

Try this. :)

https://kel.bz/post/lll/

(If you work through the prerequisites and use "understanding this post" as a sort of roadmap of what you actually need to know, this gets you about 2/3rds through undergraduate linear algebra, and you can skim through nullspaces --- all in the service of learning a generally useful tool for attacking cryptosystems).

Re: The Little Book of Linear Algebra

#53

Post by jcranmer »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations.

It's only difficult if you are wedded to a description of matrices and vectors as seas of numbers that you grind your way through without trying to instill a fuller understanding of what those numbers actually mean. The definition makes a lot more sense when you see a matrix as a description of how to convert one sense of basis vectors to another set of basis vectors, and for that, you first need to understand how vectors are described in terms of basis vectors.

Re: The Little Book of Linear Algebra

#54

Post by tptacek »

greymalik wrote: post #49
Sharlin wrote: post #22

Earlier quoted context omitted.

For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)

> there’s an incredibly motivating and rewarding way to learn the basics of linear algebra What is it?

They mean graphics programming; learning graphics programming will give you intuitions for a lot of linear algebra stuff.

Re: The Little Book of Linear Algebra

#55

Post by joshmarlow »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

The older I get the more convinced I am that "math is not hard; teaching math is hard".

Re: The Little Book of Linear Algebra

#56

Post by jameshart »

andrewla wrote: post #14

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…

What I find amazing is, given how important linear algebra is to actual practical applications, high school math still goes so deep on calculus at the expense of really covering even basic vectors and matrices.

Where vectors do come up it’s usually only Cartesian vectors for mechanics, and only basic addition, scalar multiplication and component decomposition are talked about - even dot products are likely ignored.

Re: The Little Book of Linear Algebra

#57

Post by srean »

tptacek wrote: post #52
tenacious_tuna wrote: post #51

Earlier quoted context omitted.

> I didn't think any part of linear algebra was boring. My formal linear algebra course was boring as hell, to me. The ~4 lectures my security prof dedicated to explaining just enough to do some RSA was absolutely incredible. I would pay lots of money for a hands-on what-linalg-is-useful-for course with practical examples like that.

Try this. :) https://kel.bz/post/lll/ (If you work through the prerequisites and use "understanding this post" as a sort of roadmap of what you actually need to know, this gets you about 2/3rds through undergraduate linear algebra, and you can skim through nullspaces --- all in the service of learning a generally useful tool for attacking cryptosystems).

Thanks for the notes. This is marvellous. I do not work on, or have interest in cryptography algorithms, but this is such an interesting read.

Re: The Little Book of Linear Algebra

#58

Post by eliaskickbush »

Highly recommend to anyone struggling with linear algebra to check out Linear Algebra Done Right, by Sheldon Axler. Do always keep in mind that some concepts are very verbose, but truly out of necessity. If you're talking about an N by N matrix, you're naturally going to have to distinguish N^2 different elements.

You can go very far without touching matrices, and actually find motivation on this abstract base before learning how it interops with matrices.

Re: The Little Book of Linear Algebra

#59

Post by tptacek »

srean wrote: post #57
tptacek wrote: post #52

Earlier quoted context omitted.

Try this. :) https://kel.bz/post/lll/ (If you work through the prerequisites and use "understanding this post" as a sort of roadmap of what you actually need to know, this gets you about 2/3rds through undergraduate linear algebra, and you can skim through nullspaces --- all in the service of learning a generally useful tool for attacking cryptosystems).

Thanks for the notes. This is marvellous. I do not work on, or have interest in cryptography algorithms, but this is such an interesting read.

Kelby Ludwig is such a talented explainer, it upsets me.

Re: The Little Book of Linear Algebra

#60

Post by schoen »

relaxing wrote: post #13
rw_panic0_0 wrote: post #7

how is it beginner friendly, first paragraph and already an obscure formula for non math people

I agree. I wouldn’t consider someone who has taken (and remembers) a course in set theory a beginner without some added qualifier. One of my pet peeves is using mathematical symbols beyond basic arithmetic without introducing them once by name. Trying to figure out what a symbol is and what branch of math it comes from is extremely frustrating.

There are a handful of textbooks that have a nice appendix that defines each symbol (or maybe sometimes tells you where you can go to learn more about the topic represented by the symbol!). That way they can presume that most readers are already familiar but still be helpful to those who aren't.
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