Affine-linear mapping/Functorial properties/Fact

Let ${}K$ be a field, and let ${}E,F$ and ${}G$ denote affine spaces over the vector spaces ${}U,V$ and ${}W$. Then the following statements hold.

1. The identity

   $\operatorname {Id} _{E}\colon E\longrightarrow E$

   is affine-linear.

2. The composition of affine-linear mappings

   $E\longrightarrow F$

   and

   $F\longrightarrow G$

   is again affine-linear.

3. For a bijective affine-linear mapping

   $\psi \colon E\longrightarrow F,$

   also the inverse mapping is affine-linear.

4. For ${}v\in V$, the translation

   $E\longrightarrow E,P\longmapsto P+v,$

   is affine-linear.

5. A linear mapping is affine-linear.

Proof, Write another proof