Affin-lineare mapping/Values on affine basis/Fact

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

Then, there exists a uniquely determined

affine-linear mapping

${\displaystyle \psi \colon E\longrightarrow F}$

such that

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

for all

${\displaystyle {}i\in I}$.