Endomorphism/Eigenvalue/Invariant hyperplane/Fact

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

$\varphi \colon V\longrightarrow V$

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

Then there exists a

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

of dimension ${}n-1$.

Proof, Write another proof