Affine space/Affine subspace/Characterization/Fact

< Affine space < Affine subspace < Characterization

Let ${}E$ be an affine space over the ${}K$ -vector space ${}V$ . For a subset ${}F\subseteq E$ , the following conditions are equivalent.

${}F$ is an affine subspace of ${}E$ .
For ${}P_{1},\ldots ,P_{n}\in F$ and numbers ${}a_{1},\ldots ,a_{n}$ satisfying ${}\sum _{i=1}^{n}a_{i}=1$ , we have ${}\sum _{i=1}^{n}a_{i}P_{i}\in F$ .
For two points ${}P,Q\in F$ and numbers ${}r,s\in K$ with ${}r+s=1$ , we have ${}rP+sQ\in F$ .

Proof, Write another proof