Linear mapping/Matrix and basis/Rank/Fact/Proof/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

${}\operatorname {rk} \,\varphi =\operatorname {rk} \,M\,.$

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