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 [...]