Composition of Affine Transformations is Affine Transformation

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Theorem

Let $\EE$, $\FF$ and $\GG$ be affine spaces with difference spaces $E$, $F$ and $G$ respectively.

Let $\LL: \EE \to \FF$ and $\MM: \FF \to \GG$ be affine transformations.

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


Then the composition $\MM \circ \LL: \EE \to \FF$ is an affine transformation with tangent map $M \circ L$.


Proof

Let $\NN = \MM \circ \LL : \EE \to \GG$ be the composition.

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

$\map \GG Q = \map \GG p + \map {M \circ L} {\vec {p q} }$

We find that:

\(\ds \map \GG q\) \(=\) \(\ds \map {\MM \circ \LL} q\)
\(\ds \) \(=\) \(\ds \map \MM {\map \LL p} + \map L {\vec{p q} }\) $\LL$ is an Affine Transformation

Now let:

$p' = \map \LL p$

and:

$q' = \map \LL p + \map L {\vec {p q} }$

so:

$\vec {p' q'} = \map L {\vec {p q} }$

Then:

\(\ds \map \GG q\) \(=\) \(\ds \map \MM {q'}\)
\(\ds \) \(=\) \(\ds \map \MM {p'} + \map M {\vec {p' q'} }\) $\MM$ is an Affine Transformation
\(\ds \) \(=\) \(\ds \map \MM {\map \LL p} + \map M {\map L {\vec {p q} } }\) Definitions of $p'$ and $q'$
\(\ds \) \(=\) \(\ds \map {\MM \circ \LL} p + \map {M \circ L} {\vec {p q} }\)

as required.

$\blacksquare$