Question

Suppose T:V tends to V is a linear operator. Let W be a sub space of a vector space V.Let W be invariant under the linear operator T₁:V tends to V and T₂: V tends to V.Then prove that W is also invariant under T₁+T₂ and T₁T₂.

Answer 1

Let $w\in W$ and $(T_1+T_2) (w)=T_1(w)+T_2(w).$ Let $T_1{w}=w_1\in W$ since $W$ is invariant under  $T_{1}.$ Similarly $T_2(w)=w_2\in W.$ Hence $w_1+w_2\in W$ since $W$ is a vector space.

Again $T_1T_2(w)=T_1(w_2)=w_3\in W$ since $W$ is invariant under $T_1.$ Hence we see both $T_1+T_2$ and $T_1T_2$ are invariant under $W.$