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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#31

Post by Sharlin »

andrewla wrote: post #24
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.)

To a degree I think this is true, but it requires (at least in my experience) that you have an intrinsic grasp of trigonometry for it to make sense. If you have some complex function analysis and e^itheta then you can skirt the problem for a bit, but if you're like me and have to break out soh-cah-toa whenever you break down a triangle then this method ends up being pretty tedious too.

I’m not sure what you mean. Beyond rotation matrices, there’s really only trig involved in graphics if you actively want it.

Re: The Little Book of Linear Algebra

#32

Post by asplake »

csunoser wrote: post #6

At around 7.4 orthonormal basis and there after, the tex rendering stops working on the github readme preview page. Instead, it is replaced with a red error box saying: [ Unable to render expression. ] I wonder if there is an artificial limit for the amount of latex expression that can rendered per page.

I switched to the epub at that point. Still, credit I think to github that the page renders as well as it does.

Re: The Little Book of Linear Algebra

#33

Post by andrewla »

Sharlin wrote: post #31
andrewla wrote: post #24

Earlier quoted context omitted.

To a degree I think this is true, but it requires (at least in my experience) that you have an intrinsic grasp of trigonometry for it to make sense. If you have some complex function analysis and e^itheta then you can skirt the problem for a bit, but if you're like me and have to break out soh-cah-toa whenever you break down a triangle then this method ends up being pretty tedious too.

I’m not sure what you mean. Beyond rotation matrices, there’s really only trig involved in graphics if you actively want it.

Maybe I was making unwarranted assumptions about the nature of your way to learn linear algebra. The approaches that I've seen invariably have to produce a sample matrix, and rotation is really the best example. The rotation matrix is going to have sines and cosines, and understanding what that means is not trivial; and even now if you asked me to write a rotation matrix I would have to work it out from scratch. Easy enough to do mechanically but I have no intuitions here even now.

Re: The Little Book of Linear Algebra

#34

Post by Sharlin »

cassepipe wrote: post #26
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.)

... What is this incredible motivating way ? Please do tell

Well, graphics programming itself. Learning while doing, preferably from some good resource written with graphics in mind. 2D is fine for the basics, 3D is more challenging and potentially confusing but also more rewarding.

Re: The Little Book of Linear Algebra

#35

Post by cosmic_cheese »

srean wrote: post #17
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. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these fol…

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.

Re: The Little Book of Linear Algebra

#36

Post by defanor »

Always nice to see CC-licensed textbooks. This one looks fairly minimal, not including much of explanation, illustrations, or proofs; I think those are generally useful for the initial study, but this should still work as a cheat sheet, at least.

Re: The Little Book of Linear Algebra

#37

Post by andrewla »

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 haven't looked at Strang's approach.

The "x = b / A" is a bit of a gut-punch on first look because my mind immediately tells me all the ways that that does not work. It makes a some sense once I take a second to think about it, and I can see why it would make you want to jump in a little deeper, but matrices being non-commutative makes me cringe at the idea of a division operator which does not very very clearly spell out where it appears in the chain.

Ax = b is all well and good, but AxA^-1 = bA^-1 is not meaningful; the application/composition order is very important.

Re: The Little Book of Linear Algebra

#38

Post by phforms »

WillAdams wrote: post #19
photon_lines wrote: post #11

If anyone is interested in a more visual or intuitive over-view, I made a mini-book on it as well a few years ago which you can find here: https://github.com/photonlines/Intuitive-Overview-of-Linear-...

Nice pairing the text with 3Blue1Brown's lectures on linear algebra! https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

As an autodidact who never learned this stuff at school/uni, his lectures are what made linear algebra really click for me. I can only recommend them to anyone who wants to get a visual intuition on the fundamentals of LA.

What also helped me as a visual learner was to program/setup tiny experiments in Processing[1] and GeoGebra Classic[2].

- [1] https://processing.org - [2] https://www.geogebra.org/classic

Re: The Little Book of Linear Algebra

#39

Post by dpflan »

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…

MIT OCW is an amazing resource -- anyone can learn from Strang, which is a goldmine.

He also created a course on using Linear Algebra for machine learning:

> Linear algebra concepts are key for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization–and above all a full explanation of deep learning.

- MIT OCW Course: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-an...)

- The text book website: Linear Algebra and Learning from Data (2019) https://math.mit.edu/~gs/learningfromdata/

- The Classic Linear Algebra Course: https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010...

Re: The Little Book of Linear Algebra

#40

Post by andrewla »

srean wrote: post #17
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. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these fol…

> Perhaps from graphics -- translation, rotation, reflection

Maybe ... but the fact that you included translation in the list of linear operations seems like a big red flag. Translation feels very linear but it is emphatically not [1]. This is not intended to be a personal jab; just that the intuitions of linear algebra are not easy to internalize.

Adding linear transformations is similarly scary territory. You can multiply rotations to your heart's content but adding two rotations gives you a pretty funky object that does not have any obvious intuition in graphics.

[1] I wouldn't jump into projective or affine spaces until you have the linear algebra tools to deal with them in a sane way, so this strikes me as a bit scary to approach it this way.

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