Answer
$(3,11,-8)$
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 $T(0,3,-1)=3T(0,1,0)-T(0,0,1)=3(1,3,-2)-(0,-2,2)=(3,9,-6)-(0,-2,2)=(3,11,-8)$