Affine space/Finite point set/Barycentric combinations/Barycentric combination/Exercise

Let $E$ be an affine space over a $K$-vector space $V$, and let $P_1,\dots ,P_n$ denote a finite family of points in $E$. For $j=1,\dots ,k$, let

$Q_j=\sum _{i=1}^n{a}_{ij}{P}_i,$

with $\sum _{i=1}^n{a}_{ij}=1$ for each $j$, be a family of barycentric combinations of the $P_i$. Let $b_1,\dots ,b_k\in K$ fulfilling $\sum _{j=1}^k{b}_j=1$. Show that one can write

$\sum _{j=1}^k{b}_j{Q}_j$

as a barycentric combination of the $P_i$.

Create a solution