Mathematical Sciences Department Seminar
Chris Wells, Auburn University
Monday, January 27th
4-5pm
Stratton 205
Title: Mostly orthogonal vectors
Abstract: What is the maximum number of mutually orthogonal vectors that can coexist in Rn? Well, n, of course!
What if we weaken the condition: instead of requiring that all vectors be orthogonal, let's require only that among any three of the vectors, some pair are orthogonal. How many vectors can we have now?
Observe that the union of any pair of orthogonal bases of Rn satisfies this condition, so we can have at least 2n many vectors. Can we do any better?
This question of Erdös was originally answered by Rosenfeld via a pretty tricky algebraic argument.
In this lecture, we will work through a different proof using only the Cauchy--Schwarz inequality and some basic facts from linear algebra! Afterward, we will see how these same basic ideas can be used to tackle other related problems.