Let $T$ be a linear operator on a vector space $V$, and let $c\in F$. Let $W$ be the set of eigenvectors of $T$ with eigenvalue $c$, together with $0$. Prove that $W$ is a $T$-invariant subspace.
So, I need to show $T(W)\subset W$. If I show that $\ker (T-cI)$ is $T-cI$-invariant. But $(T-cI)(v)=0\in \ker (T-cI)$, when $v\in \ker (T-cI)$. Is my argument ok? Thanks.