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Let $\phi: V \to V$ be an endomorphism over a $\mathbb{C}$-field. Show:

If there is a decomposition $V = V_1 \oplus V_2$ with $V_1,V_2 \neq \{0\}$ and $\phi$-invariant, then there exist $\phi$-invariant $U_1,U_2 \subset V$ with $\operatorname{dim} U_1 = \operatorname{dim} U_2 = 1$ and $U_1 \neq U_2$.


Since it's over $\mathbb{C}$ the minimal polynomial $\mu_\phi$ is product of linear factors (distinct?).

It is $v \in V_i \implies \phi(v) \in V_i$.


But I don't really see what to do..?

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Try to see what 1 dimensional invariant subspaces mean. And maybe make your statement stronger by requiring that $U_i \subset V_i$ which will automatically imply that $U_1 \neq U_2$.

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