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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#81

Post by _mu »

andrewla wrote: post #23
dapper_bison17 wrote: post #21

Earlier quoted context omitted.

> This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains. Which books or “non-standard” resources would you recommend t…

I have yet to encounter an approach that is not boring. You just have to power through it. This approach seems as good as any. Once you get to eigenvalues (in my opinion) things start to pick up in terms of seeing that linear spaces are actually interesting. This approach sort of betrays itself when the very first section about scalars has this line: > Vectors are often written vertically in column form, which emphas…

Maybe we can petition Paul Lockhart to do a book on Linear Algebra, I would definitely buy it.

Re: The Little Book of Linear Algebra

#82

Post by defrost »

jean_lannes wrote: post #79

As someone who took a standard undergrad linear algebra course but never really used it in my work, what are some good ways to get acquainted with practical applications of linear algebra?

There were some hints upstream: https://news.ycombinator.com/item?id=45107638

Machine learning, LLMs, RSA, etc.

It's generally useful for multivariate statistics, 3D flies (insects), in 3D space, clustering about a narrow slanting plane of light from a window slit are points that can be projected onto "the plane of best fit" - nominally the slanting plane of light.

That right there is a geometric picture of fitting a line, a plane, a lower order manifold, to a higher order data set, the errors (distance from plane), etc. and something of what Singular Value Decomposition is about (used for image enhancement, sharpening fuzzy data, etc).

The real test of applications is what kind of work do you see yourself doing? - A quick back read suggests your currently a CS student, so all unfocused potential for now (perhaps).

Re: The Little Book of Linear Algebra

#83

Post by bananaflag »

andrewla wrote: post #18
Chinjut wrote: post #15

Earlier quoted context omitted.

Why do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning. It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".

Sure, that's a way to approach it. All you have to do is stay interested in "linear functions" long enough to get there. It's totally possible -- I got there, and so did many many many other people (arguably everyone who has applied mathematics to almost any problem has). But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fac…

Yeah, concepts can make a student reject them with passion.

I remember in a differential geometry course, when we reached "curves on surfaces", I thought "what stupidity! what are the odds a curve lies exactly on a surface?"

Re: The Little Book of Linear Algebra

#84

Post by atrettel »

barrenko wrote: post #8

Tried to pick a book to get into linear algebra recently, the experience was fairly hellish. First course this, second course that, done right, done wrong... I'd to the LADR4e route, but I don't have the proof-it chops yet...

I had the same experience when I first learned linear algebra. I don't have any book recommendations, but I did want to say that for some topics, it is better to learn it by applying it than by using a book. Linear algebra was like that for me, but oddly enough I was able to learn tensor calculus from a book later (after doing a lot of problems).

Thanks to everyone recommending books too!

Re: The Little Book of Linear Algebra

#85

Post by itchingsphynx »

viewtransform wrote: post #75
cosmic_cheese wrote: post #35

Earlier quoted context omitted.

If anybody is aware of materials that teach linear algebra via graphics as suggested here, I would be interested to hear about them. As someone who learns best through practical application, maths have been by far among my greatest weak points, despite having written software for upwards of a decade. It’s limiting in some scenarios and pure imposter syndrome fuel.

3Blue1Brown [Essense of linear algebra]( https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... )

This series by Grant was very useful to review and learn and see the implications. Highly recommended.

Re: The Little Book of Linear Algebra

#86

Post by JBits »

jean_lannes wrote: post #79

As someone who took a standard undergrad linear algebra course but never really used it in my work, what are some good ways to get acquainted with practical applications of linear algebra?

A good use of linear algebra that I'm working with at the moment is the use of splines as a basis for real (vector) functions. After obtaining the matrix/vector representations you can solve for the spline coefficients (and then plot them).

Linear transforms (such as rotations and displacements) in GPU graphics.

Fourier series in signal processing.

JPEG compression.

Obtaining the best fit element in a vector space of curves given data or other constraints.

Understanding autodiff in JAX.

The mathematical definition of a tensor helps develop intuition for manipulating arrays/tensors in array libraries.

Transition matrices of a Markov chain.

PageRank.

Re: The Little Book of Linear Algebra

#88

Post by bee_rider »

jameshart wrote: post #56
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.

I think it was a brilliant and evil trick by the linear algebra folks.

Start the path at calculus. Naturally, this will lead to differential equations. Trick the engineers into defining everything in terms of differential equations.

The engineers will get really annoyed, because solving differential equations is impossible.

Then, the mathematicians swoop in with the idea of discretizing everything and using linear algebra to step through it instead. Suddenly they can justify all the million-by-millions matrices they wanted and everybody thinks they are heroes. Engineers will build the giant vector processing machines that they want.

Re: The Little Book of Linear Algebra

#89

Post by ViscountPenguin »

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 found the university level presentation of "Vector spaces -> Linear functions -> Matrices are isomorphic to linear functions" much more motivating than the rote mechanics I was taught in highschool, but it's hard to see if I would've had that appreciation without being taught the shitty way first.

Re: The Little Book of Linear Algebra

#90

Post by solarwindy »

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.

VLLMs are incredibly good at decoding math from screenshots, if you’re working from a PDF textbook. ChatGPT especially, and since it’s conversant in LaTeX, it can respond directly in the notation you don’t recognize to break it down for you. It even manages with photos of my handwritten scrawl (mostly).
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