LA23 Do all Vector Spaces have a basis? Basis via the Expansion & "two out of three" Theorems

How do we construct a basis for a Vector Space? In the previous video (   • LA22 What is Dimension? The Contraction Th...   ) we discussed the contraction theorem (start with a spanning set and throw some elements out). In this lecture, we discuss the Expansion Theorem: Any set of linearly independent vectors, in a finite dimensional vector space, can be expanded to a basis. This then shows that if a set of vectors has the same number of elements as a basis, then it spans the vector space if and only if it is linearly independent (the so-called "two out of three" Theorem). Complete Proofs presented. At the end of the lecture, we discuss (the surprisingly complicated) question of whether all vector spaces have bases. Subscribe ‪@Shahriari‬

This is a video in a series of lectures on linear algebra. The series is a rigorous treatment meant for students with no prior exposure to linear algebra. In this full undergraduate course in linear algebra, general vector spaces and linear transformations are emphasized.

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.