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This question is really basic, but I have essentially no background on algebra so I really do not know basic things. On page 4 of these notes, the author defines $\text{End}S$ to be the set of all endomorphisms of a given finite-dimensional vector space $S$. What is the definition of an endomorphism?

IamWill
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    An endomorphism is a linear transformation $S \to S$. An isomorphism between $S$ and $V$ is a bijective linear map $S \to V$. – Randall Mar 13 '21 at 15:50
  • @Randall why not make it an anwer – Kenny Lau Mar 13 '21 at 15:51
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    A homomorphism from the structure to itself. “Homomorphism” means “function that respects the structure”. In the case of vector spaces, it means “linear transformation.” So the endomorphisms of $S$ are the linear transformations $T\colon S\to S$. The isomorphisms are the invertible homomorphisms (in this case, invertible linear transformations). An endomorphism that is an isomorphism is usually called an “automorphism”. – Arturo Magidin Mar 13 '21 at 15:52
  • @KennyLau because it's probably going to just get closed or deleted. – Randall Mar 13 '21 at 15:52
  • Thank you guys! I know the question is really basic, but a search on the internet and you find a bunch of things and it is really difficult to know what is the appropriate definition for you. – IamWill Mar 13 '21 at 15:57
  • Did you mean to ask about “isomorphisms” at the end of your question, or was that meant to be “endomorphism”? Because that was a pretty sudden change of subject. – Arturo Magidin Mar 13 '21 at 16:08
  • @ArturoMagidin ops! My mistake! Corrected now! Thanks! – IamWill Mar 13 '21 at 16:10

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An endomorphism is a homomorphism $\varphi : A \to A$ with $A$ being an mathematical structure. So its a mapping from a structure to itself with respect to the structure ("not losing it through the mapping").

To be more precise and for example: In terms of $(A, \circ), (B, \ast)$ being groups or monoids with $\varphi : A \to B$:

  • $\varphi(1_A) = 1_B$ for any $1_A, 1_B$ being neutral elements in $A$ and $B$
  • $\varphi(a \circ_A b) = \varphi(a) \ast_B \varphi(b)$ for all $a \in A$ and $b \in B$
Algebruh
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