Affine space/Finite point family/Affinely independent/Definition

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

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

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

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

wit

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

is only possible if

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

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