Linear mapping/Matrix for basis/Surjective and column generating system/Exercise

Let $K$ be a field and let $V$ and $W$ be vector spaces over $K$ of dimensions $n$ and $m$. Let

$\phi :V⟶W$

be a linear map, described by the matrix $M\in {\mathrm{Mat}}_{m\times n}(K)$ with respect to two bases. Prove that $\phi$ is surjective if and only if the columns of the matrix form a system of generators for $K^m$.

To solution