TITLE: What is the essence of Kernel or Null Space?
QUESTION [5 upvotes]: I describe Kernel in this way: Kernel(In group theory, Linear Algebra, etc.) measures the degree to which a map or morphism fails to be injective. In "finite dimensional" case, it also measures the degree to which a map or morphism fails to be surjective.
How to give a more general definition of Kernel, instead of just defining them on a special Algebraic structure, such as groups?

REPLY [2 votes]: What’s not mentioned yet: given  a homomorphism $f\colon G\to H$ between two groups (or vectorspaces) then one may establish a new group $G/\ker(f)$ via equivalence relation.  Prominent example: $f\colon\mathbb Z\to\mathbb Z$, $x\mapsto2x$. Then $\ker(f)=2\mathbb Z$ so $f$ produces the famous $\mathbb Z/2\mathbb Z=\mathbb Z_2$.