Let $k$ be a field, let $m \geq 0$ be an integer, let $A$ be an [affine space](/page/Affine%20Space) modeled on a $k$-[vector space](/page/Vector%20Space) $V$, and let $p_0,\ldots,p_m \in A$. Assume that $p_0,\ldots,p_m$ are affinely independent, equivalently that the displacement vectors $p_1-p_0,\ldots,p_m-p_0$ are linearly independent in $V$. Then for every point

\begin{align*} q \in \operatorname{Aff}\{p_0,\ldots,p_m\}, \end{align*}

there exists a unique tuple $(\lambda_0,\ldots,\lambda_m) \in k^{m+1}$ such that

\begin{align*} \lambda_0+\cdots+\lambda_m=1 \end{align*}

and

\begin{align*} q=\lambda_0p_0+\cdots+\lambda_mp_m, \end{align*}

where the right-hand side is interpreted as the affine combination of the points $p_0,\ldots,p_m$ with coefficients $\lambda_0,\ldots,\lambda_m$.