Let φ : V → V {\displaystyle {}\varphi \colon V\rightarrow V} be a linear mapping, and let V = U ⊕ W {\displaystyle {}V=U\oplus W} be a direct sum of φ {\displaystyle {}\varphi } -invariant linear subspaces. Show that φ {\displaystyle {}\varphi } is nilpotent if and only if φ | U {\displaystyle {}\varphi {|}_{U}} and φ | W {\displaystyle {}\varphi {|}_{W}} are nilpotent.
