Endomorphism/Eigenvalue/Invariant hyperplane/Fact

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}n}$-dimensional vector space. Let

${\displaystyle \varphi \colon V\longrightarrow V}$

be a linear mapping, and let ${\displaystyle {}\lambda \in K}$ be an eigenvalue of ${\displaystyle {}\varphi }$.

Then there exists a

${\displaystyle {}\varphi }$-invariant linear subspace ${\displaystyle {}U\subset V}$

of dimension ${\displaystyle {}n-1}$.