Characterization of Affine Transformations

Theorem
Let $\mathcal E$ and $\mathcal F$ be affine spaces over a field $k$.

Let $\mathcal L: \mathcal E \to \mathcal F$ be a mapping.

Then $\mathcal L$ is an affine transformation for all points $p, q \in \mathcal E$ and all $\lambda \in k$:


 * $\mathcal L \left({\lambda p + \left({1 - \lambda}\right) q}\right) = \lambda \mathcal L \left({p}\right) + \left({1 - \lambda}\right) \mathcal L \left({q}\right)$

where $\lambda p + \left({1 - \lambda}\right) q$ and $\lambda \mathcal L \left({p}\right) + \left({1 - \lambda}\right) \mathcal L \left({q}\right)$ denote barycenters.

Sufficient Condition
Let $\mathcal L$ be an affine transformation.

Let $L$ be the tangent map.

Let $r \in \mathcal E$ be any point.

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

Thus we find:

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


 * $\mathcal L \left({\lambda p + \left({1 - \lambda}\right) q}\right) = \lambda \mathcal L \left({p}\right) + \left({1 - \lambda}\right) \mathcal L \left({q}\right)$

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

Fix a point $p \in \mathcal E$, and define for all $u \in E$:
 * $L\left(u\right) = \mathcal L\left(p + u\right) - \mathcal L\left(p\right)$

Let $q = p + u$.

Then:
 * $\mathcal L \left({q}\right) = \mathcal L \left({p}\right) + L \left({u}\right)$

So to show that $\mathcal L$ 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 $u, v \in E$:
 * $L \left({\lambda u}\right) = \lambda L \left({u}\right)$

and:
 * $L \left({u + v}\right) = L \left({u}\right) + L \left({v}\right)$

First of all:

Now it is to be shown that
 * $L \left({u + v}\right) = L \left({u}\right) + \left({v}\right)$

First:
 * $p + u + v = \dfrac 1 2 \left({p + 2 u}\right) + \dfrac 1 2 \left({p + 2 v}\right)$

Now:

From the above calculation:
 * $L \left({u + v}\right) = \mathcal L \left({p + u + v}\right) - \mathcal L \left({p}\right) = L \left({u}\right) + L \left({v}\right)$

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