Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 2 / b / iii

For $v\in V_{\lambda }$, use commutation to obtainThus $Sv\in V_{\lambda }$, proving that every $T$-eigenspace is an $S$-invariant subspace. This does not assert that an arbitrary eigenvector of $T$ is already an eigenvector of $S$.