Linear mapping/Dimension formula/Simple cases/Remark

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

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

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

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

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

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

and

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