Endomorphism/Eigenvalue/Invariant hyperplane/Fact/Proof

Proof

Because of the condition and fact, the mapping ${}\varphi -\lambda \operatorname {Id} _{V}$ has a nontrival kernel. Hence, this mapping is not injective and, due to fact, also not surjective. Therefore,

${}B:=\operatorname {Im} {\left(\varphi -\lambda \operatorname {Id} _{V}\right)}\subset V\,$

is a strict linear subspace of ${}V$. It follows that there exists also a linear subspace ${}U\subset V$ of dimension ${}n-1$, which contains ${}B$. For ${}u\in U$, we have

${}\varphi (u)=\lambda u+(\varphi -\lambda \operatorname {Id} _{V})u\in U+B\subseteq U\,.$

Hence, the image of ${}U$ belongs to ${}U$, that is, ${}U$ is ${}\varphi$-invariant.

To fact