Let E {\displaystyle {}E} be an affine space over a K {\displaystyle {}K} -vector space V {\displaystyle {}V} , and let P 1 , … , P n {\displaystyle {}P_{1},\ldots ,P_{n}} denote a finite family of points in E {\displaystyle {}E} . For j = 1 , … , k {\displaystyle {}j=1,\ldots ,k} , let
with ∑ i = 1 n a i j = 1 {\displaystyle {}\sum _{i=1}^{n}a_{ij}=1} for each j {\displaystyle {}j} , be a family of barycentric combinations of the P i {\displaystyle {}P_{i}} . Let b 1 , … , b k ∈ K {\displaystyle {}b_{1},\ldots ,b_{k}\in K} fulfilling ∑ j = 1 k b j = 1 {\displaystyle {}\sum _{j=1}^{k}b_{j}=1} . Show that one can write
as a barycentric combination of the P i {\displaystyle {}P_{i}} .
be an
affine space
over a
-vector space
, and let
denote a finite family of points in . For
,
let
for each 
, be a family of
barycentric combinations
of the 
. Let
fulfilling
.
Show that one can write
.
