In my Algebra class, my professor defines the kernel in the context of group homomorphisms, but this seems completely unrelated to the normal definition. Is these equivalent to the normal definition? It looks completely different.
Let $G, H$ be groups and $u, v: G \to H$ be homomorphisms of groups. We define the pair $(K, i)$ to be the kernel of the pair $(u, v)$ provided $i: K \to G$ is a homomorphism such that $u \circ i = v \circ i$ and, for any group $F$ and homomorphism $w : F \to G$ such that $u \circ w = v \circ w$, there is a unique homomorphism $w' : F \to K$ such that $w = i \circ w'$.
EDIT: One commenter (now deleted) suggested that this is the category theoretic definition of the kernel. It looks to me that the definition given on Wikipedia is still slightly different than the definition given to me in class. The Wikipedia definition defines the kernel of a some morphism $f$, while my professor defines the kernel of a pair $(u, v)$.