Set of complementary subspaces as an affine space

Question

Let $V$ be a vector space and $W$ a fixed vector subspace of $V$. Denote by $\mathcal{E}$ the set of vector subspaces $U$ of $V$ such that $V=U\oplus W$. It is stated in the Examples section here that $\mathcal{E}$ is an affine space over $\mbox{Hom}(V/W,W)$. My question is: What is the free transitive action of $\mbox{Hom}(V/W,W)$ on $\mathcal{E}$?

This can be equivalently formulated as follows: Consider a short exact sequence of vector spaces $$ 0\longrightarrow W \longrightarrow V \longrightarrow U \longrightarrow 0, $$ then the set of splittings of this sequence is an affine space over $\mbox{Hom}(U,W)$. Again, how?

Answer 1

For an $f:V/W\to W$ and a $U\in\mathcal E$ define $$U+f\ :=\ \{u+f([u]_W)\,:\,u\in U\}$$ where $[u]_W$ is the coset $u+W$ in $V/W$.