Composition of Affine Transformations is Affine Transformation

Theorem

Let $\mathcal E$, $\mathcal F$ and $\mathcal G$ be affine spaces with difference spaces $E$, $F$ and $G$ respectively.

Let $\mathcal L: \mathcal E \to \mathcal F$ and $\mathcal M: \mathcal F \to \mathcal G$ be affine transformations.

Let $L$ and $M$ be the tangent maps of $\mathcal L$ and $\mathcal M$ respectively.

Then the composition $\mathcal M \circ \mathcal L: \mathcal E \to \mathcal F$ is an affine transformation with tangent map $M \circ L$.

Proof

Let $\mathcal N = \mathcal M \circ \mathcal L : \mathcal E \to \mathcal G$ be the composition.

We want to show that for any $p, q \in \mathcal E$

$\mathcal G \left({q}\right) = \mathcal G \left({p}\right) + M \circ L \left({\vec{p q} }\right)$

We find that:

 $\displaystyle \mathcal G \left({q}\right)$ $=$ $\displaystyle \mathcal M \circ \mathcal L \left({q}\right)$ $\displaystyle$ $=$ $\displaystyle \mathcal M \left({\mathcal L \left({p}\right)}\right) + L \left({\vec{p q} }\right)$ $\mathcal L$ is an Affine Transformation

Now let:

$p' = \mathcal L \left({p}\right)$

and:

$q' = \mathcal L \left({p}\right) + L \left({\vec{p q} }\right)$

so:

$\vec{p' q'} = L \left({\vec{p q} }\right)$

Then:

 $\displaystyle \mathcal G \left({q}\right)$ $=$ $\displaystyle \mathcal M \left({q'}\right)$ $\displaystyle$ $=$ $\displaystyle \mathcal M \left({p'}\right) + M \left({\vec{p' q'} }\right)$ $\mathcal M$ is an Affine Transformation $\displaystyle$ $=$ $\displaystyle \mathcal M \left({\mathcal L \left({p}\right)}\right) + M \left({L \left({\vec{p q} }\right)}\right)$ Definitions of $p'$ and $q'$ $\displaystyle$ $=$ $\displaystyle \mathcal M \circ \mathcal L \left({p}\right) + M \circ L \left({\vec{p q} }\right)$

as required.

$\blacksquare$