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

$\varphi \colon V\longrightarrow W$

be a linear mapping.

a) Show that ${}\varphi$ is surjective if and only if there exists a linear mapping

$\psi \colon W\longrightarrow V$

such that

${}\varphi \circ \psi =\operatorname {Id} _{W}\,.$

b) Let now ${}\varphi$ be surjective, and set

${}S={\left\{\psi :W\rightarrow V\mid \psi {\text{ linear}},\,\varphi \circ \psi =\operatorname {Id} _{W}\right\}}\,,$

and let ${}\psi _{0}\in S$ be fixed. Define a bijection between ${}\operatorname {Hom} _{K}{\left(W,\operatorname {kern} \varphi \right)}$ and ${}S$, such that ${}0$ is mapped to ${}\psi _{0}$.

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