Jordan matrix/No direct sum decomposition/Exercise

Let

$\varphi \colon V\longrightarrow V$

be an endomorphism that is described, with respect to a suitable basis, by a ${}n\times n$-Jordan matrix. Show that there does not exist a direct sum decomposition

${}V=U\oplus W\,$

into ${}\varphi$-invariant linear subspaces ${}U,W\subset V$.

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