28

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.

Lorenzo B.
  • 2,252
EngDR
  • 493

2 Answers2

41

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.

Types of mappings between sets

Lorenzo B.
  • 2,252
Adam Marsh
  • 721
  • I'm new to math.stackexchange, can someone explain the downvote? I found that I could not embed the image without more reputation, but if I could it would have been self-explanatory I think. – Adam Marsh Feb 01 '18 at 03:47
  • Added more details to perhaps address the downvote. Also, I cannot comment on the above answer, but as @al-jebr mentions, an endomorphism need not be injective. – Adam Marsh Feb 01 '18 at 19:47
  • 4
    “For any algebraic structure... epimorphism” careful. Epimorphism means right-cancellable, and while every surjective homomorphism is an epimorphism, there are nonsurjective epimorphisms, e.g., the embedding $\mathbb{Z}\hookrightarrow\mathbb{Q}$ is an epimorphism of rings (and of multiplicative semigroups). Monomorphism, left cancellable is almost always equivalent to injective, but not always. These are true for vector spaces and linear algebras, but saying “for any algebraic structure” is too general. – Arturo Magidin Nov 10 '20 at 01:07
  • Thanks Arturo, added a note that in categories besides vector spaces the category theoretic definitions may be distinct. – Adam Marsh Nov 11 '20 at 20:28
4

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 $(H,\star)=(G,\cdot)$.

MoebiusCorzer
  • 3,284
  • 1
    Dear Corzer thanks for the relpy, can you explain what does {0_v1} means – EngDR Nov 03 '15 at 19:13
  • I use the notation $0_{V_{1}}$ to denote the null vector from the vector space $V_{1}$. For example, if $V_{1}=\mathbb{R}^{2}$, $0_{V_{1}}=(0,0)$. If $V_{1}=\mathbb{R}^{5}$, $0_{V_{1}}=(0,0,0,0,0)$. – MoebiusCorzer Nov 03 '15 at 23:21
  • 4
    Endomorphisms are not necessarily injective. That is a monomorphism. Endomorphisms are maps from a mathematical object to itself. – user5826 Nov 16 '17 at 02:04