Answer
a) $(-14,-7)$
b) $(1,1,z)$
Work Step by Step
We know that
$T(v_1,v_2,v_3)=(4v_2-v1,4v_1+5v_2)$
Thus
$T(2,−3,-1)=(4(-3)-2,4(2)+5(-3))=(−14,-7)$
Let $(x,y,z)$ be the pre-image of $(3,9)$. Then $T(x,y,z)=(4y-x,4x+5y)=(3,9)$.
Thus $4y-x=3,4x+5y=9$. From this we get $x=1,y=1, z=z.$
