Affine space/Affine basis/Hyperplane realization/Fact

${\displaystyle E\longrightarrow K^{n+1},P\longmapsto (a_{1},\ldots ,a_{n},a_{n+1}),}$

where ${\displaystyle {}a_{i}}$ denotes the barycentric coordinates of ${\displaystyle {}P}$, is an affine-linear mapping, which provides an affine isomorphism between ${\displaystyle {}E}$ and the affine subspace ${\displaystyle {}F\subset K^{n+1}}$, guven by

${\displaystyle {}F={\left\{x\in K^{n+1}\mid \sum _{i=1}^{n+1}x_{i}=1\right\}}\,.}$

The translating vector space of ${\displaystyle {}F}$ is

${\displaystyle {}W={\left\{x\in K^{n+1}\mid \sum _{i=1}^{n+1}x_{i}=0\right\}}\,.}$