Affine space/Finite point family/Affinely independent/Definition

Affinely independent

Let ${}E$ be an affine space over a ${}K$-vector space ${}V$, and let

$P_{1},\ldots ,P_{n}$

be a finite family of points in ${}E$. We say that this family of points is affinely independent, if an equality

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

wit

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

is only possible if

${}a_{i}=b_{i}\,$

for all ${}i=1,\ldots ,n$.