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

   ${\mathrm{Id}}_E:E⟶E$

   is affine-linear.

2. The composition of affine-linear mappings

   $E⟶F$

   and

   $F⟶G$

   is again affine-linear.

3. For a bijective affine-linear mapping

   $\psi :E⟶F,$

   also the inverse mapping is affine-linear.

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

   $E⟶E,P⟼P+v,$

   is affine-linear.

5. A linear mapping is affine-linear.

Proof, Write another proof