Affine space/Affine basis/Hyperplane realization/Fact/Proof

Proof

Due to fact, there exists a uniquely determined affine-linear mapping

$E\longrightarrow K^{n+1}$

sending ${}P_{i}$ to the ${}i$-th standard vector ${}e_{i}$. Because of fact, the point

${}P=\sum _{i=1}^{n+1}a_{i}P_{i}\,$

is sent to

$\sum _{1=1}^{n+1}a_{i}e_{i}.$

Because of

${}\sum _{i=1}^{n+1}a_{i}=1\,,$

this image point belongs to ${}F$. Bijectivity is clear.

To fact