vector space epimorphism splits

Let \(K\) be a field, \(U,V\) vector spaces and \(f: V \twoheadrightarrow U\) be a vectorspace-epimorphism
Then \(f\) is split epi, i.e. there exists a vector space homomorphism \(\iota: U \rightarrow V\) such that

\begin{align*} f \circ \iota = \mathrm{id}_U \end{align*}

Choose a basis \(u_i\) of \(U\) (cf. Existenz einer Basis in einem Vektorraum).
Then since \(f: V \twoheadrightarrow U\) is surjective, we may find preimages \(v_i\) of \(u_i\), i.e.

\begin{align*} f(v_i) = u_i \end{align*}

Then define

\begin{align*} g: U \rightarrow& V \\ u_i \mapsto& v_i \end{align*}

via extending (cf. Existenz und Eindeutigkeit eines Homomorphismus durch die Abbildungen der Basis)

It follows that

\begin{align*} f(g(u_i)) = u_i \end{align*}

hence \(f \circ g\) is the identity on the basis \(u_i\).
Hence by uniqueness of the extension, it is the identity on \(U\)