Linear mapping/Matrix and basis/Rank/Fact

Let ${}K$ denote a field, and let ${}V$ and ${}W$ denote ${}K$-vector spaces of dimensions ${}n$ and ${}m$. Let

$\varphi \colon V\longrightarrow W$

be a linear mapping which is described by the matrix ${}M\in \operatorname {Mat} _{m\times n}(K)$, with respect to bases of the spaces.

Then

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

holds.

Write a proof