Linear mapping/On finite dimensional space/Rank/Definition

Let ${\displaystyle {}K}$ denote a field, let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ denote ${\displaystyle {}K}$-vector spaces, and let

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

denote a ${\displaystyle {}K}$-linear mapping. Suppose that ${\displaystyle {}V}$ has finite dimension. Then we call

${\displaystyle {}\operatorname {rk} \,\varphi$ :=\dim _{K}{\left(\operatorname {Im} \varphi \right)}\,}

the rank of ${\displaystyle {}\varphi }$.