Answer
a) $(2,-4)$
b) $(4,4)$
Work Step by Step
a)
We know that $T(v_1,v_2)=(v1,v_1+2v_2)$
Thus $T(2,−3)=(2,2+2(-3))=(2,-4)$
b)
Let $(x,y)$ be the pre-image of $(4,12)$. Then $T(x,y)=(x,x+2y)=(4,12)$.
Thus $x=4$, $x+2y=12\\4+2y=12\\2y=8\\y=4$. From this we get $x=4,y=4.$
So the pre-image is (4,4).
