$U$ and $V$ are vector spaces, their Cartesian product is defined by
$$U\times V = [(u,v): u \in U, v \in V]$$
addition and multiplication defined by
$$(u_1,v_1) + (u_2,v_2) = (u_1+u_2, v_1+v_2)$$ $$r(u,v) = (ru,rv)$$
Let the $T_1: V \rightarrow W$ and $T_2:V\rightarrow W$ be linear transformations, and define a transformation $S: V \times V \rightarrow W \times W$ by
$$S[(v_1,v_2)] = (T_1(v_1),T_2(v_2)$$
how do I go about proving the linear transformation here? I know the idea behind proving linear transformations for composition when they are $T_1(T_2(v_1))$ for example but I am not sure how to prove it in this case.
To prove addition I have gotten this far but I don't know where to go from here:
$$S[(v_1,v_2)+(v_3,v_4)] = S[(v_1,v_2)] + S[(v_3,v_4)]$$ $$=[T_1(v_1),T_2(v_2)] + [T_1(v_3),T_2(v_4)]$$
\begin{align*} S[(v_{1},v_{2})+(v_{3},v_{4})]&=S(v_{1}+v_{3},v_{2}+v_{4})\\ &=(T_{1}(v_{1}+v_{3}),T_{2}(v_{2}+v_{4}))\\ &=(T_{1}(v_{1})+T_{1}(v_{3}),T_{2}(v_{2})+T_{2}(v_{4}))\\ &=(T_{1}(v_{1}),T_{2}(v_{2}))+(T_{1}(v_{3}),T_{2}(v_{4}))\\ &=S(v_{1},v_{2})+S(v_{3},v_{4}). \end{align*}