TITLE: Difference between epimorphism, isomorphism, endomorphism and automorphism (with examples)
QUESTION [25 upvotes]: Can somebody please explain me the difference between linear transformations such as epimorphism, isomorphism, endomorphism or automorphism?
I would appreciate if somebody can explain the idea with examples or guide to some good source to clear the concept.

REPLY [27 votes]: For any algebraic structure, a homomorphism preserves the structure, and some types of homomorphisms are:

Epimorphism: a homomorphism that is surjective (AKA onto)
Monomorphism: a homomorphism that is injective (AKA one-to-one, 1-1, or univalent)
Isomorphism: a homomorphism that is bijective (AKA 1-1 and onto); isomorphic objects are equivalent, but perhaps defined in different ways
Endomorphism: a homomorphism from an object to itself
Automorphism: a bijective endomorphism (an isomorphism from an object onto itself, essentially just a re-labeling of elements)

Note that these are common definitions in abstract algebra; in category theory, morphisms have generalized definitions which can in some cases be distinct from these (but are identical in the category of vector spaces).
So a linear transformation $A\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ is a homomorphism since it preserves the vector space structure (vector addition, scalar addition and multiplication, scalar multiplication of vectors), e.g. $A(av+w)=aA(v)+Aw$. It is an epimorphism if its image is $\mathbb{R}^{m}$, a monomorphism if it has zero kernel, an endomorphism if $n=m$, and an automorphism (as well as an isomorphism) if all of these are true.
The below figure might be helpful. More details here.