Affin-lineare mapping/Values on affine basis/Fact

Determination theorem for affine mappings

Let ${}K$ be a field, and let ${}E$ and ${}F$ denote affine spaces over the vector spaces ${}V$ and ${}W$. Let ${}P_{i}$, ${}i\in I$, denote an affine basis of ${}E$, and let ${}Q_{i}$, ${}i\in I$, denote a family of points in ${}F$.

Then, there exists a uniquely determined

affine-linear mapping

$\psi \colon E\longrightarrow F$

such that

${}\psi (P_{i})=Q_{i}\,$

for all

${}i\in I$.

Proof, Write another proof