Matrix/Surjective/Right inverse/Exercise

Let ${}M$ be an ${}m\times n$-matrix and ${}\varphi \colon K^{n}\rightarrow K^{m}$ the corresponding linear mapping. Show that ${}\varphi$ is surjective if and only if there exists an ${}n\times m$-matrix ${}A$ such that

${}M\circ A=E_{m}$.

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