Characterization of Affine Transformations

Theorem
Let $\EE$ and $\FF$ be affine spaces over a field $K$.

Let $\LL: \EE \to \FF$ be a mapping.

Then $\LL$ is an affine transformation :


 * $\forall p, q \in \EE: \forall \lambda \in K: \map \LL {\lambda p + \paren {1 - \lambda} q} = \lambda \map \LL p + \paren {1 - \lambda} \map \LL q$

where $\lambda p + \paren {1 - \lambda} q$ and $\lambda \map \LL p + \paren {1 - \lambda} \map \LL q$ denote barycenters.

Sufficient Condition
Let $\LL$ be an affine transformation.

Let $L$ be the tangent map.

Let $r \in \EE$ be any point.

Then by definition we have:
 * $\lambda p + \paren {1 - \lambda} q = r + \lambda \vec{r p} + \paren {1 - \lambda} \vec{r q}$

Thus we find:

Necessary Condition
Suppose that for all points $p, q \in \EE$ and all $\lambda \in \R$:


 * $\map \LL {\lambda p + \paren {1 - \lambda} q} = \lambda \map \LL p + \paren {1 - \lambda} \map \LL q$

Let $E$ be the difference space of $\EE$.

Fix a point $p \in \EE$, and define for all $u \in E$:
 * $\map L u = \map \LL {p + u} - \map \LL p$

Let $q = p + u$.

Then:
 * $\map \LL q = \map \LL p + \map L u$

So to show that $\LL$ is affine, we are required to prove that $L$ is linear.

That is, we want to show that for all $\lambda \in k$ and all $u, v \in E$:
 * $\map L {\lambda u} = \lambda \map L u$

and:
 * $\map L {u + v} = \map L u + \map L v$

First of all:

Now it is to be shown that
 * $\map L {u + v} = \map L u + \map L v$

First:
 * $p + u + v = \dfrac 1 2 \paren {p + 2 u} + \dfrac 1 2 \paren {p + 2 v}$

Now:

From the above calculation:
 * $\map L {u + v} = \map \LL {p + u + v} - \map \LL p = \map L u + \map L v$

This shows that $L$ is linear, and therefore concludes the proof.