use function to find (a) the image of v and (b) the preimage of w:-

Question

1-T(v₁,v₂)=(v₁+v₂,v₁-v₂),v=(3,-14),w=(3,19)

2-T(v₁,v₂,v₃)=(v₂-v₁,v₁+v₂,2v₁),v=(2,3,0),w=(-11,-1,10)

Answer 1

To find the image of $v$ all you have to do is pass $v$ to the input of $T$ as

1) $T(v1,v2) = T(3,-14) = (3-14,3+14) = (-11,17) $

2) $T(v1,v2) = T(2,3,0) =(3-2,2+3,2(2)) = (1,5,4) $

To find the pre-image of w is a bit more challenging as you need to figure out which input produces $w$

1)so say for $ w= (3,19)$ you have to find which v gives

v1+v2 = 3

v1-v2= -14

Add both equations to get v1 as 2v1 = -11 so v1 = -5.5 and v2 = 3-v1=3+5.5=8.5. So we can say that the preimage of w is (-5.5,8.5)

2) Same methodology could be followed to get v1,v2,v3 that satisfy

v2-v1=-11

v1+v2=-1

2v1 = 10

From the last equation v1 = 5 hence from the second equation we get that v2 = -1 - v1 = -6 so note that we have solved the equation with v3 so for any v3 the preimage is (5,-6,v3)