Affine independence

Remarks

Example: in $\R^2$, the points $(0,0)$, $(1,0)$, $(0,1)$ are affinely independent, since the vectors $(1,0)$ and $(0,1)$ are linearly independent. Non-example: the points $(0,0)$, $(1,1)$, $(2,2)$ are not, since $(2,2) = 2 \cdot (1,1)$. The condition does not depend on which point plays the role of $p_0$. An equivalent form: for all $\lambda_0, \dots, \lambda_k \in F$ with $\sum_i \lambda_i = 0$ and $\sum_i \lambda_i (p_i - p_0) = 0$, all $\lambda_i$ are $0$. Another equivalent form: no $p_j$ is an affine combination of the other points. In an affine space of dimension $n$, at most $n + 1$ points can be affinely independent.