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 if and only if 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:
 * $\lambda p + \left({1 - \lambda}\right) q = r + \lambda \vec{ rp } + \left(1 - \lambda\right) \vec{ rq }$

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 $u \in E$:
 * $L\left(u\right) = \mathcal L\left(p + u\right) - \mathcal L\left(p\right)$

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

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 let us show that
 * $L\left({ u + v }\right) = L\left({ u }\right) + \left({ v }\right)$

We first write:
 * $\displaystyle p + u + v = \frac 12 \left({ p + 2u }\right) + \frac 12 \left({ p + 2v }\right)$

Now we compute

Now from this calculation, we conclude that:
 * $L\left({ u + v }\right) = \mathcal L\left({ p + u + v }\right) - \mathcal L\left({ p }\right) = L\left({ u }\right) + L\left({ v }\right)$

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