Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 300: 1

$(11,-8)$

We know that $T(v_1,v_2)=(v_1+v_2,v_1-v_2)$ Thus $T(3,-4)=(3+(-4),3-(-4))=(-1,7)$ Let $(x,y)$ be the pre-image of $(3,19)$. Then $T(x,y)=(x+y,x-y)=(3,19).$ Thus $x+y=3,x-y=19$ From this we get $x=11,y=-8$. So the pre-image of $(3,19)$ is $(11,-8)$