Answer
See below
Work Step by Step
Let $v, w \in V$ and $k \in R$ scalar.
$$T(v+w)=((u_1,v+w);(u_2,v+w))\\
=((u_1,v)+(u_1,w);(u_2,v)+(u_2,w))\\
=((u_1,v),(u_2,v))+((u_1,w)+(u_2,w))\\
=T(v)+T(w)$$
$$T(kv)=((u_1,kv);(u_2,kv))\\
=(k(u_1,v);k(u_2,v)) \\
=k((u_1,v);(u_2,v))
=kT(v)$$
It is a linear transformation.