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The Little Book of Linear Algebra

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Re: The Little Book of Linear Algebra

#41

Post by bmacho »

Sharlin wrote: post #22
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…

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 is no such thing as affine algebra: https://en.wikipedia.org/wiki/Affine_algebra

Re: The Little Book of Linear Algebra

#42

Post by bmacho »

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

Linear algebra has motivations and applications everywhere, since its main defining concepts, 'addition' and 'transformation that keeps sums' are everywhere. So a linear algebra curse is a huge pile of disjointed facts. It is not such a set of material that can have motivations behind it.

But the good news is that if you are only interested in for example geometry, game theory, systems of linear equations, polynomials, statistics, etc, then you can skip 80% of the content of linear algebra books. You don't have to read them, understand them, memorize them. You'll interact with a tiny part of linear algebra anyway, and you don't have to do that upfront.

Re: The Little Book of Linear Algebra

#43

Post by srean »

andrewla wrote: post #40
srean wrote: post #17

Earlier quoted context omitted.

> 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 m…

Mea culpa about translation.

For a moment I was thinking in homogeneous coordinates - that's not the right thing to do in the introductory phase.

Thanks for catching the error and making an important point. I am letting my original comment stand unedited so that your point stands.

About rotations though, one need not let the cat out of the bag and explain what addition of rotation is *.

One simply defines addition of two linear operators as the addition of the vectors that each would have individually produced. This can be demonstrated geometrically with arrows, without fixing coordinates.

* In 2D it's a scaled rotation.

Re: The Little Book of Linear Algebra

#44

Post by Sharlin »

andrewla wrote: post #33
Sharlin wrote: post #31

Earlier quoted context omitted.

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…

Rotation matrices are somewhat mysterious to the uninitiated, but so is matrix multiplication until it "clicks". Whether it ever clicks is a function of the quality of the learning resource (I certainly do not recommend trying to learn linalg via 3D graphics by just dabbling without a good graphics-oriented textbook or tutorial – that usually doesn’t end well).

Anyway, I believe that it's perfectly possible to explain rotation matrices so that it "clicks" with a high probability, as long as you understand the basic fact that (cos a, sin a) is the point that you get when you rotate the point (1, 0) by angle a counter-clockwise about the origin (that's basically their definition!) Involving triangles at all is fully optional.

Re: The Little Book of Linear Algebra

#45

Post by te »

cosmic_cheese wrote: post #35
srean wrote: post #17

Earlier quoted context omitted.

> 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.

codingthematrix.com

Re: The Little Book of Linear Algebra

#46

Post by Sharlin »

bmacho wrote: post #41
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 is no such thing as affine algebra: https://en.wikipedia.org/wiki/Affine_algebra

There are affine spaces, and there is an algebra of the elements of affine spaces. That is, rules that describe how the elements can be manipulated. There are affine transforms, affine combinations, affine bases, and so on, all of them analogous to the corresponding concepts in linear algebra.

(The term "algebra" can also refer to a particular type of algebraic structure in math, but that’s not what I meant.)

Re: The Little Book of Linear Algebra

#47

Post by [deleted] »

andrewla wrote: post #37
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 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 sp…

[deleted]

Re: The Little Book of Linear Algebra

#48

Post by srean »

andrewla wrote: post #33
Sharlin wrote: post #31

Earlier quoted context omitted.

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…

In 2D there's an alternative. One can rotate purely synthetically, by that I mean with compass and straight edge. This avoids getting into transcendentals.

Of course I am not suggesting building synthetic graphics engines :) but the synthetic approach is sufficient to show that the operation is linear.

Re: The Little Book of Linear Algebra

#49

Post by greymalik »

Sharlin wrote: post #22
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…

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?

Re: The Little Book of Linear Algebra

#50

Post by rossant »

A linear algebra course without graphics? When I learnt it at school almost 25 years ago, the teacher made schematics all the time to explain the visual intuition behind each concept. I was totally confused when he introduced the abstract definition of a vector space with the addition and scalar multiplication. Then he drew some arrows. Then it all suddenly made sense.
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