Linear surjective mapping/Linear section/As affine-linear space/Exercise

Let $K$ be a field, let $V$ and $W$ be finite-dimensional $K$-vector spaces, and let

$\phi :V⟶W$

be a linear mapping.

a) Show that $\phi$ is surjective if and only if there exists a linear mapping

$\psi :W⟶V$

such that

$\phi \circ \psi ={\mathrm{Id}}_W.$

b) Let now $\phi$ be surjective, and set

$S=\{\psi :W\to V\mid \psi \text{ linear},\phi \circ \psi ={\mathrm{Id}}_W\},$

and let ${\psi }_0\in S$ be fixed. Define a bijection between ${\mathrm{Hom}}_K(W,\mathrm{kern}\phi )$ and $S$, such that $0$ is mapped to ${\psi }_0$.

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