Answer
See below.
Work Step by Step
In this case the transformation is linear if $T(v_1+v_2)=T(v_1)+T(v_2)$ and $T(av_1)=aT(v_1)$ for any $v_1,v_2$ in the domain and any $a$ scalar.
Hence, here
$T(1,1)=0.5T(2,0)+\frac{1}{3}T(0,3)=0.5(1,1)+\frac{1}{3}(3,3)=(0.5,0.5)+(1,1)=(1.5,1.5)$
$T(0,1)=0T(2,0)+\frac{1}{3}T(0,3)=0(1,1)+\frac{1}{3}(3,3)=(0,0)+(1,1)=(1,1)$
