trace: what is it really?
[02-17-23]
a better definition of the trace of a linear operator (which some people like to draw as matrices relative to arbitrary bases, sad)
group products
[02-16-23]
Ever wondered what a semi-direct group product is? what, if anything is the difference between an inner semidirect product and an outer semidirect product? Yeah. I've wondered these things too. But then I met $\Z_5 \rtimes \Z_4$. And it became less mysterious.
Bela Fleck Theorem
[12-12-23]
The Beck Fiala theorem is a fundamental result in discrepancy theory. Basically it says, if you have a collection of subsets of a universe and each element of the universe shows up in a limited number of the sets in your collection then there is a 2-coloring of the universe such that all sets have a roughly equal number of red and blue elements.
Cauchy Shwarz almost equality
[06-25-23]
This is part of my series of doing problems from the book "problems from the book" (by Titu Adnreescu Gabriel Dospinescu). Today is about an interesting variant of Cauchy Schwarz.
the power of linear-dependence
[12-19-23]
One of the most basic facts of linear algebra is that whenever you have $n+1$ vectors in an $n$-dimensional vectors space, there must be one of your vectors which can be expressed as a linear combination of the other vectors. This seemingly simple fact can prove some really interesting combinatorial statements.
