Minimal polynomial/Homothety/Example

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

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

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

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