Let $\alpha$ be a linear operator on a vector space $V$, and supoose that $V$ is $\alpha$-cyclic, say generated by $v\in V$. Suppose further that $V=U_1\bigoplus U_2$ for non-trivial $\alpha$-invariant subspaces $U_1$ and $U_2$ of V.
Prove that both $U_1$ and $U_2$ are $\alpha$-cyclic.
Suppose further that $\alpha$ has a minimal polynomial $m_\alpha(x)$. Prove that $$m_\alpha(x)=m_{\alpha|_{U_1}}(x)m_{\alpha|_{U_2}}(x).$$
I tried to argue that $U_1$ ($U_2$) is generated by the term of the smallest degree but could not show that either is contained by the cyclic subspace generated. For part 2, I feel like it is intended that we show $m_{\alpha|_{U_1}}(x),m_{\alpha|_{U_2}}(x)$ are coprime do not know how to proceed. Any hints will be appreciated.
I guess you are in a finite dimensional setting. For part 1: Let $v = u_1 + u_2$ with $u_1 \in U_1, u_2 \in U_2$ I claim that $U_1$ is generated by $u_1, \alpha(u_1), \alpha ² (u_1) ... .$ Let $u \in U_1.$ Since $u \in V$ we can write $$ u = \sum\limits_i c_i \alpha^i (v) = \sum\limits_i c_i \alpha^i(u_1 + u_2) = \sum\limits_i c_i \alpha^i (u_1) + \sum\limits_i c_i \alpha^i (u_2). $$ Since the sum $V = U_1 \oplus U_2$ is direct you have $u = \sum\limits_i c_i \alpha^i (u_1).$ Same for $U_2$.
For 2: choose a basis for $U_1$ and $U_2$. The union forms a basis for $V$. Now check the matrix representation of $\alpha$ with respect to this basis.