Affine space/Affine basis/Hyperplane realization/Fact

Let ${}K$ be a field, and let ${}E$ denote an affine space with an affine basis ${}P_{1},\ldots ,P_{n},P_{n+1}$.

Then the mapping

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

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

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

The translating vector space of ${}F$ is

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

Proof, Write another proof