Difference between epimorphism, isomorphism, endomorphism and automorphism (with examples) 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.
 A: In linear algebra, an epimorphism between vector spaces is a surjective linear application $A:V_{1}\to V_{2}$, that is $\text{Im}(A)=V_{2}$. For example, $B:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto x+y$.
In linear algebra, an endomorphism from a vector space to itself. For example, $B:\mathbb{R}\to\mathbb{R}:x\mapsto 2x$.
Edited (30/08/2018): removed 'injective' and corrected the definition.
In linear algebra, an isomorphism between vector spaces is a both surjective and injective linear application $A:V_{1}\to V_{2}$, that is $\text{Ker}(A)=\{0_{V_{1}}\}$ and $\text{Im}(A)=V_{2}$. 
An automorphism is an isomorphism between a vector space and itself. For example, $B:\mathbb{R}\to\mathbb{R}:x\mapsto x$.
In a more general setting, a morphism $\phi$ between two groups $(G,\cdot)$ and $(H,\star)$ is an application $G\to H:g\mapsto \phi(g)$ such that, for all $g,g'\in G$, we have $\phi(g\cdot g')=\phi(g)\star\phi(g')$ and such that $\phi(e_{G})=e_{H}$ where $e_{I}$ is the identity element of $I=G,H$.
An epimorphism between such two groups is a surjective morphism $\phi:G\to H$, i.e. for any $h\in H$, there exists $g\in G$ such that $h=\phi(g)$.
An endomorphism between such two groups is an injective morphism $\phi:G\to H$, i.e. for all $g,g'\in G$ with $\phi(g)=\phi(g')$, it implies $g=g'$.
An isomorphism between two such groups is a both injective and surjective morphism.
An automorphism is an isomorphism with $(H,\star)=(G,\cdot)$.
A: 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.

