Proof of the Day: The Kernel’s Injective Fried Chicken

Let be a linear map between vector spaces. Show that T is injective if and only if .
Proof:
Suppose first that T is injective. Then implies , so .
Now suppose . Then implies which implies , and . So, T is injective.

Proof of the Day: Homomorphisms and Abelian Groups

Let be a homomorphism of Abelian groups.  Show that for each .
Proof:
Since f is a homomorphism, and since , an Abelian group, then:

(by associativity)
(f is a homomorphism)

The last line is true because (but that’s another proof).

Proof of the Day (POD): Rings, Cancellation, and Zero Divisors

I’m going to try and start a new feature here, the “proof of the Day”.  It’s based on the crossfit idea of the Workout of the Day where they post a new workout everyday.  In this case, I’ll try (and maybe some of my colleagues here will also) and post a new proof (not EVERY [...]

Busy-ness! And a love for Maple…

The quarter is off to a running start! I wish I could post more here, but… Whew!  Hopefully this post will make up for my absence!
So here’s a little potpourri for you:

Basic Math: Dumb as a Monkey

New study shows that even college students can perform as well as monkeys on an arithmetic test.
The results indicate that monkeys perform approximate mental addition in a manner that is remarkably similar to the performance of the college students. These findings support the argument that humans and nonhuman primates share a cognitive system for nonverbal [...]

Let’s Study Linear Algebra

For anyone interested in getting a jump ahead for MTH 444/544: Advance Linear/Multilinear Algebra, we’ll be meeting at 10 a.m. on Monday, 17 December in the Neuberger Atrium. Read the first four sections of the text (up through page 33) and be ready to discuss all the homework problems. That is – [...]

“Concrete Groupoid Theory”, “Axiomatic Theories”: the Same Subject?

Here’s Part I
And Part II

What are Tannakian Categories?

And why are they important?
But for me (at least), the real reason why Tannakian categories are interesting is because of the role they play in Grothendieck’s original conception of motives.

The Lyceum Mathematikoi Test Post

This is a test post.  This post is only a test.  Do not pass go, do not collect two-hundred dollars mod n.