Eigenvectors of commuting linear transformations

#linear_algebra

Let $S : V \to V$ and $T : V \to V$ be linear transformations that commute, i.e. $S \circ T = T \circ S$ . Let $v \in V$ be an eigenvector of $S$ such that $T (v) \neq 0$ . Prove that $T (v)$ is also an eigenvector of $S$ .

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Let $S : V \to V$ and $T : V \to V$ be linear transformations that commute, i.e. $S \circ T = T \circ S$. Let $v \in V$ be an eigenvector of $S$ such that $T(v) \ne 0$. Prove that $T(v)$ is also an eigenvector of $S$.