Affine space/Affine Basis/Barycentric coordinates/Fact

Theorem about barycentric coordinates

Let ${}P_{i}$, ${}i\in I$, denote an affine basis in an affine space ${}E$ over the ${}K$-vector space ${}V$.

Then, for every point

${}P\in E$, there exists a unique barycentric representation

${}P=\sum _{i\in I}a_{i}P_{i}\,.$

Proof, Write another proof