= Affine independence
{title2=$\sum_i\lambda_i p_i=0,\ \sum_i\lambda_i=0\ \Longrightarrow\ \lambda_i=0$}
= Affinely independent
{synonym}
Points $p_0,\ldots,p_k$ in an <affine space> are <affinely independent> if the only affine dependence displayed above has all coefficients zero, using any choice of origin to read the <vector> equation. Since the coefficients sum to zero, replacing the origin leaves that equation unchanged. Substituting $\lambda_0=-\sum_{i=1}^k\lambda_i$ shows that this condition is equivalent to <linear independence> of $p_1-p_0,\ldots,p_k-p_0$. It also says that an affine combination of the points has unique coefficients. In $\mathbb R^n$, at most $n+1$ points can be <affinely independent>; $n+1$ such points span a full-dimensional <simplex>.
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