Vectors Are Not Lists of Numbers (Part 1) - From Zero to Geo 1.8

In this video, we explore the idea of the uniqueness of coefficients in linear combinations, which leads us to the idea of components. While components can be thought of as lists of numbers, it is vital to distinguish between a vector and its components. We also explore some of the computational applications that components allow for. 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 Next video here: https://www.youtube.com/watch?v=If8r7vo4uoU Discord: https://discord.gg/3Zj59zA2Rg Patreon: https://www.patreon.com/sudgylacmoe Patreon Supporters: AxisAngles Christoph Kovacs David Johnston p11 Richard Penner Rosario Sections: 00:00 Introduction 00:36 Uniqueness of Coefficients 01:43 Argument for Uniqueness 03:05 Argument for Non-uniqueness 03:33 Condition for Uniqueness 04:08 Argument for Uniqueness in 3D 05:44 Components 07:32 Vector Are Not Lists of Numbers 09:01 Building Better Bases 10:33 Notation for the Standard Basis 11:52 Simple Exercise 12:43 Formulas for Vector Operations 13:14 Formula for Length in 2D 13:58 Formula for Length in 3D 15:35 Exercises 16:09 Conclusion