quality:correction, stylistic:tone, linguistic:paraphrasing
Endomorphisms

An affine transformation or endomorphism of an affine space A {\displaystyle A} is an affine map from that space to itself. One important family of examples is the translations: given a vector v → {\displaystyle {\overrightarrow {v}}}, the translation map T v →: A → A {\displaystyle T_{\overrightarrow {v}}:A\rightarrow A} that sends a ↦ a + v → {\displaystyle a\mapsto a+{\overrightarrow {v}}} for every a {\displaystyle a} in A {\displaystyle A} is an affine map. Another important family of examples are the linear maps centred at an origin: given a point b {\displaystyle b} and a linear map M {\displaystyle M}, one may define an affine map L M, b: A → A {\displaystyle L_{M,b}:A\rightarrow A} by L M, b (a) = b + M (a − b) {\displaystyle L_{M,b}(a)=b+M(a-b)} for every a {\displaystyle a} in A {\displaystyle A}.

After making a choice of origin b {\displaystyle b}, any affine map may be written uniquely as a combination of a translation and a linear map centred at b {\displaystyle b}.