Affine-linear mapping/Barycentric combination/Fact

Let ${}E$ and ${}F$ denote affine spaces over a field ${}K$, and let

$\psi \colon E\longrightarrow F$

denote a mapping.

Then, ${}\psi$ is

affine-linear if and only if for every barycentric combination ${}\sum _{i\in I}a_{i}P_{i}$ with ${}P_{i}\in E$, the equality

${}\psi {\left(\sum _{i\in I}a_{i}P_{i}\right)}=\sum _{i\in I}a_{i}\psi (P_{i})\,$

holds.

Proof, Write another proof