Minimal polynomial/Homothety/Example

For the identity ${\displaystyle {}\operatorname {Id} _{V}}$ on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, the minimal polynomial is just ${\displaystyle {}X-1}$. This polynomial is sent under the evaluation homomorphism to

${\displaystyle {}\operatorname {Id} _{V}-\operatorname {Id} _{V}=0\,.}$

A constant polynomial ${\displaystyle {}a_{0}}$ is sent to ${\displaystyle {}a_{0}\operatorname {Id} }$, which is not, with the exception of ${\displaystyle {}a_{0}=0}$ or ${\displaystyle {}V=0}$, the zero mapping.

For a homothety, that is, a mapping of the form ${\displaystyle {}\lambda \operatorname {Id} _{V}}$, the minimal polynomial is ${\displaystyle {}X-\lambda }$, under the condition ${\displaystyle {}\lambda \neq 0}$ and ${\displaystyle {}V\neq 0}$. For the zero mapping on ${\displaystyle {}V\neq 0}$, the minimal polynomial is ${\displaystyle {}X}$, in case ${\displaystyle {}V=0}$, it is the constant polynomial ${\displaystyle {}1}$.