An Overview of the Operations in Geometric Algebra

This video is an overview of some of the basic operations in geometric algebra. It covers operations like the geometric, outer, regressive, and inner products, some of the involutions, and a few other operations. It also tries to describe these operations in a way that applies to arbitrary flavors of geometric algebra, to allow for a smoother transition into these other flavors.

Patreon Supporters:
AxisAngles
Bill Six
Christoph Kovacs
David Johnston
LoganMP
p11
Richard Penner
Rosario

Sections:
00:00 Introduction
01:09 Multivectors
03:31 Grade Projection
04:50 Geometric Product
05:12 Geometric Understanding of the Geometric Product
06:40 Algebraic Understanding of the Geometric Product
10:01 Understanding the Geometric Product with Transformations
12:01 Reverse
14:21 Grade Involution
15:38 Magnitude
18:54 Outer Product
22:54 Regressive Product
26:51 Dual
30:33 Inner Product
39:49 Conclusion

[JackDespero]

As a physicist, my frustration with this videos is that I see the amazing potential of these tools, but I have other math so ingrained in my brain that I fail to use these methods for any physical problem beyond toy cases.
Honestly, it would be great if you could show us how to do it, although that might be out of your wheelhouse.
In anway, amazing videos. I always love them and learn so much from them.

[jperez]

this is excellent. this is probably the second or third time i watched this while slowly working on the textbook of macdonald

[evandrofilipe]

Ah man I've been wanting this for so long, so glad you're still doing this.

Edit: its literally like you knew I read the wiki article on geometric algebra yesterday, perfect timing

[kikivoorburg]

At first I was somewhat concerned about the use of the dagger symbol to refer to the reverse of the multivector since that symbol is already used for the Hermitian conjugate of a complex matrix. However, you can construct a complex-matrix basis for 3D VGA using the pauli matrices and it turns out that taking the Hermitian conjugate of those matrices behaves exactly like the reverse operation! Mathematics is wonderful!!

P.s. Your videos on GA have been amazing for me, it's the first bit of mathematics I've explored on my own without any educational institution guiding my hand and it's been one of the best experiences in my life so far! Possibly the most amazing moment was when I accidentally rederived the pauli matrices (actually a set of matrices equivalent to the pauli ones but multiplied by some constants) by first trying to find a 2D VGA basis using 2x2 matrices (I already knew the identity matrix and matrix for 'i' so just filled in the two remaining degrees of freedom), and then expanding that to 3D VGA with 2x2 complex matrices. My conception of the matrices (which I'd met in the context of spinor operators in QM) immediately went from them being esotheric and confusing to intuitive and elegant. As happens with many of these sorts of YouTube series, it seems people are tailing off a little as time goes on, but I hope you don't lose the motivation to keep making these videos as they are invaluable resources to those of us that have stuck around! I'm a student at the moment so likely won't be able to support on Patreon any time soon, but I would certainly like to and will do so when I am in a financially stable position!

[CubOfJudahsLion]

These definitions are clearer than anything in the literature I own, and that includes material by Hestenes, Dorst and Vince. I'm not exaggerating. And of course having universal definitions makes this video a treasure trove.

[TheSummoner]

It would be great if you could make a video (I think even a short one could suffice) about one of the best yet most overlooked cases one can make for the use of geometric algebra over regular vector algebra or complex Clifford algebras, namely the fact that if you allow real linear transformations to have multivectors as their eigenspaces you gain a geometric (and real) interpretation for what would otherwise be a collection of complex eigenvectors and eigenvalues which make no sense other than from an algebraic standpoint.

[emjizone]

This video ease a lot the implementation of general geometric algebra in any programming language. Thank you for that.

[tunabilgin]

I'm amazed by the shear amount of high quality maths channels these days. I simply can't keep up!

[PeeterJoot]

Really like the way you covered the hodge dual construction in this video. I've avoided most use of it, out of fear that I didn't understand it, but it has now lost it's mystery.

[dipfish]

Your video is a pleasure for the eyes and for the mind.

[AndrewBrownK]

oh I'm so happy! I've been looking forward to the next installment!

[AllemandInstable]

this is the clearest video on Geometric Algebra you have released yet
when you are already familiar with vector spaces, it is really easy to follow and understand the difference between the very familiar vector space and the powerful geometric algebra

[Bolpat]

Wow, this and the STA video really push the limits of my understanding. I’m quite convinced that if this weren’t a video where I can skip back and rewind a segment my attention dropped or I didn’t feel having understood it, there’s no chance I’d have learned something in less than 45 minutes.
It’s so stupid how inefficient reading a book or hearing a lecture live is for getting a basic understanding of concepts. Both has worked for me, sure, but I see the clear difference in pace. I learned complex analysis basically only through live lecture, learned forcing with textbook The only advantage a live lecture has is I can interrupt the lecturer and ask questions in real time. Watching a recorded video, I can pause any time to think about something, rewind for whatever reason, I can take screenshots, pause to write a detailed question about what is on screen with a timestamp so that you, the creator, has it as easy as it gets to write an answer.

[outofthebots]

Thank you for taking the time to make these videos. I am studying at Uni atm and couldn't quite reconcile in my head much of the math using cross products that lead me to research exactly how they work. I ended up on your YouTube channel and now everything makes so much more sense.

[caiocysneiros]

You're doing an amazing job, making hard things really accessible. Thanks a lot! Keep up the good work!

[gbnam]

let's go new geometric algebra content

i've been somewhat struggling with the different kinds of product recently, so this is actually perfect.

[torstenmiertsch]

Your videos on Geometric Algebra are, in my opinion, very good for gaining a deep look and a better understanding of what complex numbers, quaternions, etc. are all about and what their geometric interpretation actually is. In this context, I would have a few suggestions. Could you also go into the topic of dual quaternions, i.e. the combination of quaternions and dual numbers? How can this be represented with the help of geometric algebra? And furthermore, the topic of dual numbers or, more generally, the geometry of the generalized complex numbers. The classical complex numbers are so-called elliptic complex numbers, the dual numbers are the parabolic complex numbers and so-called hyperbolic complex numbers are described.

[getoff-pqpe]

Absolutely fantastic. I get so many thoughts and ideas from this. Thank you so much!

[SillySussySally]

this and the introduction are in the hall of fame of my videos on math.

[sophiaelementaris]

Oh, I've been waiting so long for this video