Affin-lineare mapping/Values on affine basis/Fact/Proof

Proof

Let ${}i_{0}\in I$. Due to fact, there exists a uniquely determined linear mapping

$\varphi \colon V\longrightarrow W$

such that

${}\varphi ({\overrightarrow {P_{i_{0}}P_{i}}})={\overrightarrow {Q_{i_{0}}Q_{i}}}\,$

for all ${}i\in I\setminus \{i_{0}\}$. Therefore,

${}\psi (R)=Q_{i_{0}}+\varphi ({\overrightarrow {P_{i_{0}}R}})\,$

is an affine-linear mapping with the properties looked for. Note that

${}\psi (P_{i_{0}})=Q_{i_{0}}+\varphi ({\overrightarrow {P_{i_{0}}P_{i_{0}}}})=Q_{i_{0}}\,$

and

${}\psi (P_{i})=Q_{i_{0}}+\varphi ({\overrightarrow {P_{i_{0}}P_{i}}})=Q_{i_{0}}+{\overrightarrow {Q_{i_{0}}Q_{i}}}=Q_{i}\,$

holds. Such an affine mapping ${}\psi$ is uniquely determined by its linear part and the image of just one point, so that

${}\psi _{0}=\varphi \,$

must hold.

To fact