Linear mapping/Dimension formula/Simple cases/Remark

Let ${\displaystyle {}\varphi \colon V\rightarrow W}$ be a linear mapping, where ${\displaystyle {}V}$ has finite dimension. The dimension formula can be illustrated with the following special cases. If ${\displaystyle {}\varphi }$ is the zero mapping, then ${\displaystyle {}\operatorname {kern} \varphi =V}$ and

${\displaystyle {}\operatorname {rk} \,\varphi =0\,.}$

If ${\displaystyle {}\varphi }$ is injective, then ${\displaystyle {}\operatorname {kern} \varphi =0}$, and

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

The rank is always between ${\displaystyle {}0}$ and the dimension of the source space ${\displaystyle {}V}$. If ${\displaystyle {}\varphi }$ is surjective, then

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

and

${\displaystyle {}\dim _{K}{\left(V\right)}=\dim _{K}{\left(\operatorname {kern} \varphi \right)}+\dim _{K}{\left(W\right)}\,.}$