The Basis Theorem
Channel: Andrew Misseldine
Date: 2022-03-07

In this video, we state and prove the Basis Theorem, that is, for all vector spaces each basis has the same cardinality. This implies that the dimension of a vector space is well-defined. We do this using Zorn's Lemma. We also state and prove the Underdetermined and Overdetermined Theorems as corollaries of the Basis Theorem, which state that a set larger than the dimension of a vector space must be linearly independent and a set smaller than the dimension of a vector space cannot span. We then discuss some of the concerns one must be aware of when working with infinite-dimensional vector spaces.

This is lecture 24 (part 1/1) of the lecture series offered by Dr. Andrew Misseldine for the course Math 4230 - Abstract Algebra II at Southern Utah University. A transcript of this lecture can be found at Dr. Misseldine's website or through his Google Drive at: 
https://drive.google.com/file/d/1Ss4sXxkGa0A9_dOjmEr9OSisTyNg0bnu/view

This lecture is based upon Sections 20.3 of Abstract Algebra: Theory and Applications (http://abstract.ups.edu/) by Tom Judson. Please post any questions you might have below in the comment field and Dr. Misseldine (or other commenters) can answer them for you. Please also subscribe for further updates.