$T\colon V \rightarrow V$ is a linear transformation. IF $U_1$ and $U_2$ are invariant subspaces of $V$ under $T$, prove $U_1 + U_2$ invariant under $T$.
I supposed it is invariant under $T$. Then proved that the element in $U_1 + U_2$ is also in $U_1$ and in $U_2$. So, they are invariant, correct?
Please help :)
By definition $U_i$ invariant under $T$ means $T(U_i) \subseteq U_i$, for $i=1,2$. Now take $u_1+u_2 \in U_1+U_2$. By linearity $$T(u_1+u_2) = T(u_1)+T(u_2) \in U_1+U_2,$$ since $T(u_i) \in U_i$ by assumption. This is true for any $u_1+u_2 \in U_1+U_2$, thus $T(U_1+U_2) \subseteq U_1+U_2$, namely $U_1+U_2$ is invariant under $T$.