Let $T$ be a linear operator on a finite dimensional vector space $V$ and $W$ be a $T$-invariant subspace of $V$.Suppose $v_1,v_2,...,v_n$ are the eigenvectors corresponding to distinct eigenvalues $\lambda_1,...,\lambda_n$. Suppose $v_1+v_2+...+v_n\in W$ and prove that $v_1,v_2,...,v_n\in W$.
SOLUTION APPROACH: I have taken an approach as $W$ is $T$ invariant and $v_1+v_2+...+v_n\in W$. This clearly implies $$T(v_1+v_2+...+v_n)=\lambda_1v_1+...+\lambda_n v_n\in W.$$ I am stuck here.I can't conclude anything else from here. Thanks in advance.If I've done some mistake please do forgive me.