You can add scalars and vectors! From Zero to Geo 1.11

In this video, I show that contrary to popular belief, you can add scalars and vectors! While this particular idea isn't terribly useful, with a bit of generalization it can lead the way to many useful things, such as multivectors.

Sorry for how long this video took to make! While the final product isn't that long, I rewrote parts of it so many times. I even removed a significant section at the last moment because it was too confusing. I hope that in the end, it was worth it and that this video is informative.

This video is a part of "From Zero to Geo", a series where we formulate geometric algebra, an incredibly powerful branch of mathematics, from the ground up. Full playlist here: https://www.youtube.com/playlist?list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy

Discord: https://discord.gg/3Zj59zA2Rg

Patreon: https://www.patreon.com/sudgylacmoe

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
trb

Sections:
00:00 Introduction
00:39 Questions
01:14 Analogy with basis vectors
02:16 Exercise
02:48 Paravectors as a linear space
03:12 Zeros
03:58 General direct sums
05:10 Properties of direct sums
06:11 Conclusion
06:40 Rigorous Construction