Affine-linear mapping/Barycentric combination/Fact/Proof

Proof

Let and denote the vector spaces for and for , respectively. Suppose first that is affine-linear with linear part

Let a barycentric combination with and be given. Then we have (with an arbitrary point )

Now, suppose that the mapping is compatible with barycentric combinations. We set

for , where is any point. We first show that this is independent of the chosen point . The sum

is a barycentric combination of the point , see exercise. Therefore, we have in the equality

Hence, we have in the equality

and, therefore,

We have to show that is linear. For and , we have

Thus, we have