Linear mapping/Matrix and basis/Rank/Fact

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

${\displaystyle \varphi \colon V\longrightarrow W}$

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