Why Geometric Algebra Should be in the Standard Linear Algebra Curriculum by Logan Lim

Abstract:

Geometric algebra is an extension of R^n that includes as special cases: The complex numbers, quaternions, exterior algebras, dual numbers, split-complex numbers, dual quaternions and more! When applied to multivariable calculus, it generalizes the fundamental theorem of calculus on manifolds to include the divergence theorem, curl theorem, and gradient theorem, and as a result Green’s and Stoke’s theorem, as special cases of a single statement. It also simplifies many geometric operations in computer graphics by eliminating the need for matrices in projections, rotations, and reflections. Though we can only cover the ‘main idea’ of geometric algebra in the allotted time, this talk will be a buffet of ideas you can explore for this fascinating and deceptively simple algebraic object.

#MTBoS #geometricalgebra #bivector #quaternions #complexnumbers #grassmann #clifford

[curtishorn]

There was a series of Physics textbooks that used geometric algebra instead of traditional vector calculus published many years ago. It would be nice if there were some math books that built up to that and went in parallel.

[Michallote]

I completely advocate for this as well. With that powerful of a system you could cover so much more of the courses because complex ideas simplify a lot. The main takeaway is that key geometric concepts become very intuitive.

About the implementation of courses. I remember using the cross product since highschool. And most classmates actually found it very unintuitive as well. We should completely drop the concept in my opinion, in favour of the wedge operation or the outer product. Then at undergraduate level analytic geometry and linear algebra courses should adopt this framework, and then the usual multivariable calculus should become multivector calculus. Although matrices should still have their place. They are incredibly useful for computations. And computers are nowadays extremely efficient and well built around them. I still don't know how would you incorporate current continuum mechanics to be worked around this style of notation. Tensor notation is incredibly compact already, I don't see many people talking about their similarities between GA as often as complex numbers and quaternions do

[jackpeterkettley]

ABSOLUTELY AGREE, as a recent compsci grad who took many courses in LA, i agree, and would have muched preferred this, i thought my classes were stale, and I think geometric algebra is a really good framework and context for solving some of the most pressing problems across multiple fields in science.

[dreastonbikrain]

36:24 you said it right the first time, i.e.: R_{n, 0, 1} is projective geometric algebra (with the caviat that the dual space is which we care about hence: R_{n, 0, 1}*. Spacetime algebra (the non projective one) has signature R_{1, 3, 0}. If I recall correctly the new shiny projective spacetime algebra has signature R_{3, 1, 1}. All of this assuming that the signature means {+, -, 0}.

[dreastonbikrain]

14:08 Uuuh shiny new notation idea which I somehow did not think about for dealing with graded parts of multivectors, ... I will steal that sir - thank you!

[MathwithMing]

[Solved] I'm lost at 9:32, is "u wedge v" a real number (as it can be added to u dot v, the number given by the dot product)? In that case isn't the uv (the left hand side) just a number as well? OR should the + sign understood purely as a simple rather than the usual addition between real numbers?

[Edit] I figured it out (after watching Alan Macdonald's playlist, cited in the latter part of this video): the plus sign in the definition of geometric product
uv = u dot v + u wedge v
is understood purely symbolically, instead of the the usual addition in R. That is, uv is an ordered pair (u dot v, u wedge v), not a real number. The plus sign is merely a suggestive notation to facilitate computation in a more natural manner. Just like with complex numbers a+bi, we can still do complex arithmetic without the symbols + and i but they make things easier to memorize.

Thanks for the presentation. It is really eye opening!

[ricardodelzealandia]

Irrespective of curriculum, where on the internet is there a course that teaches geometric algebra from beginning to end? I can't find anything and I can't find any schools teaching it locally (Sydney, Oz). I need a standard uni-like program with exams and course work.

[wstomv]

Great talk (unfortunately with some writos, i.e. mistakes in the scribbling, though they should cause no confusion if you pay attention and trust your own judgment). In particular, I remember, e_1 e_2 + e_1 e_2 + e_3 e_1 (instead of e_1 e_2 + e_2 e_3 + e_3 e_1; occurring twice), and u v + u v (instead of u v + v u); similarly u v - u v (instead of u v - v u), and near the end when mentioning projection, the u_parallel and u_perpendicular were switched in the diagram.

[morgengabe]

I think they should can the linear algebra course and do:
algebra 1 - intro to linear/nonlinear algebra; work on set theory and associative/commutative sequences
algebra 2 - combinatorial methods/proof writing; groups
algebra 3 - structural analysis; the heirarchy of algebraic structures

[BlueGiant]

I'm a bit puzzled as to why at 32:24 the partial derivative of time and the del/nabla operator were shown rather than de/nablal as the vector derivative operator (which has a history of using either del or a box).

[KipIngram]

I *could not* agree with you more. I think geometric algebra brings a unity and clarity to things that we just don't get from the standard way all this stuff is taught. I think teaching us cross products instead of bivectors is... well, it's offensive. It's like saying "All you ever need to know is the stuff you can do with this hack." The cross product *only works* in 3D. Even when we use it in 2D we're really "cheating" - those 2D cross products fall outside of our "space at hand." We're expected to just hum and overlook this fact.

[yorailevi]

Good lecture, a question;
with everything matrix it's easy to make a computer do numerical calculations, how can geometric algebra can be used to do the same calculations?

[rrrbb]

my opinion on teaching it: don't use standard notation e_1, e_2, etc. use:

up = -down
right = -left

parallel same directions in space square to 1.
up up = 1. right right = 1.
perpendicular directions anti-commute.
right up = -up right.

counter-clock rotation is obvious, and creates anti-commute:
right up = up left = left down = down right.

don't even worry or think about dot or wedge product. use forward/backward to get to 3D. you will reproduce complex numbers, vectors, circular trig, quaternions immediately. i think you can go a long time only dealing with directions in space before dealing with objects that square to 1, or square to 0.

[PaulMurrayCanberra]

18:25 so, i is a bivector (confusing, because I is the unit n-vector). what does it mean to raise a scalar to the power of a blade? You know what - I'll ask chatgpt.

[vtrandal]

At 9:25 you want to introduce us to the driving force to all of this (i.e. the Clifford-Grassman product), but you really don't. You show a couple of equations with justification and keep moving as though that was supposed to make sense.

[PaulMurrayCanberra]

2:57 - "And importantly, the subspaces must be (mumble)". Damn. I bet this is going to be a problem in a minute.

[tissuepaper]

okay, I gotta say, somebody needs to fix the terminology being used in higher math. "blade" is a stupid name like "magma", "ring", etc. that isn't descriptive (meaning you have to simply memorize the definition instead of having the name as a mnemonic), and the fact that "k-vector" and "multivector" refer to different types of objects is awful. same goes for the distinction between a "k-blade" and "n-blade". It was hard enough to learn to use complex numbers without a pile of stupid, obfuscated names to memorize. it's a hard sell for a teacher whose students really only need to be able to solve these things by hand in 2D.

[caspermadlener]

This video does a kinda bad job at communicating why geometric algebra should be teached.

Showing the effectiveness of a certain method requires the following two steps:
1. Explain how the existing methods deal with a problem that you can solve with your method.
2. If your method is doing the same thing as an existing method, explain why your method still gives a more complete and general view of the solution, and how it impacts generalisations.

For example, if you want to explain why umbral calculus (look it up, it is pretty cool) should be teached, you should show how it solves linear differential equations and linear recurrence relations in the same way as the laplacian and the generating series, but it unifies these two concepts in an algebraically rigorous way, and actually gives deeper insight in these solutions.

The worst way to do this is to say "oh look at how nice we can write this", without actually showing how they would work, or just implying that it simplifies some of the proofs (like with the determinant).

You wouldn't tell people how food tasted if you have samples of that food, right? This isn't any different.
At least you didn't just gave definitions of Clifford algebra, that would be like giving out empty plates :)

[alphalunamare]

This has totally failed to explain for why.

[rv]

"Geometric algebra" is just a goofy way of saying "some relatively elementary geometric aspects of Clifford algebras". It hardly deserves the name of "theory". It's stuff like the determinant trick to compute curl in vector calculus.