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

$\varphi \colon V\longrightarrow W$

be a linear map, described by the matrix ${}M\in \operatorname {Mat} _{m\times n}(K)$

with respect to two bases. Prove that ${}\varphi$ is surjective if and only if the columns of the matrix form a system of generators for ${}K^{m}$.

To solution