Prove Affine Transformation is a sum of Linear Transformation and Translation from axioms

Question

Background

Affine Transformation acting on vectors is usually defined as the sum of a linear transformation and a translation (especially in some CS books). i.e.,

For vector spaces $V, W$, $f: V \rightarrow W$ is affine if
$f(\vec{v}) = L(\vec{v})+\vec{b}$.
where $L: V \rightarrow W$ is a linear transformation, and $\vec{b} \in W$ is a fixed vector.

Then, it can be shown that $f$ preserves affine combinations. That is,
for all affine combination $\sum_i \lambda_iu_i$ (where $\sum_i \lambda_i$ = 1),
$f(\sum_i \lambda_iu_i)$ = $\sum_i \lambda_i f(u_i)$

Question

My question is, how to prove the converse? Namely, prove that
for any $f$ that perserves affine combinations, it can be expressed as a sum of a linear transformation and a translation.

I have read that it is true, but I haven't succeeded in finding or coming up with a proof. Thank you in advance.

Answer 1

Hint: Assume a map $f:V\to W$ preserves affine combinations, then consider $L(v):=f(v)-f(0)$ and prove that it's linear.
Thus with $b=f(0)$ we do have $f(v)=L(v)+b$.