Nilpotents & Real Numbers | Intro to Geometric Algebra

This video introduces the basics of a nilpotent (null vector) extension of the Real Numbers. The goal of this video is to help develop the intuition necessary for the essentials of Geometric Algebra.
// Patreon
patreon.com/Eccentric282
// Timestamps
00:00 - The Canonical Nilpotent Basis
01:00 - Defining a New Number
01:39 - Even/Odd Geometric Number Decomposition
01:52 - New Conjugations
03:05 - Inverses
03:34 - Extra Decompositions
03:59 - Outro
//Music
Music by Vincent Rubinetti
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
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[@leotimm6805]

So good ! Loved your video !
I tried to do the math at 1:27 for myself (prove that the matrix representation of geomtric numbers agrees with the geometric multiplication) and I was amased to see the row and column vector combine.
How did you come up with the row vector [ba, a] and the column vector [ba b] ? And is there a natural way to generalize it to more dimentions ? If we started with 3 nilpotenent vectors a, b, c such that ab + ba = ac + ca = bc + cb = 1 whould we have obtained 3D spacetime algebra ?

[@EccentricTuber]

Thank you for the feedback! It always is such a great feeling when the math just works!
While I'd like to take credit for the ideas in this video, I sadly didn't discover them. I first read about them in "Matrix Gateway to Geometric Algebra, Spacetime and Spinors" by Garret Sobczyk (Chapter 3). As for their construction, roughly speaking it comes from longterm experience with matrix mechanics: you just kinda know that's how it's done. I know, it's not a good explanation for a mathematician lol.
The book discusses nilpotent constructions for the Spacetime Algebra (I'm pretty sure), but they're drastically less introductory compared to the (1+1)D spacetime. Hope this helps!