"just ignore the new wording"
You can say T(x) = Ax
If T goes from R^n to R^m, (let''s say m>n). see this example of n=2 m=3 and finding A.
T(e1)={5,-7,2}, T(e2)={-3,8,0}, x={x1,x2}, = x1e1+x2e2 and T(x)=x1[T(e1)+x2T(e2) or ...
x1{5,-7,2}+x2{-3,8,0}=
5x1-3x2
-7x1+8x2
2x1+0
If you want it "invertible" or reverse solved, A must be an invertible matrix (made of elemetary matrices or row operations, meaning "it can solve" without free variable or other issues).
You shouldn't be mystified by it. You are just multiplying, say a A a 2x3 matrix by a, say, x is a 1x2 and by rule ending up with a 3x3. the 3x3 may be "onto" or "one to one" (which depends on A)
Ax=T(x) = blah, a 3x3 matrix. simple multiplication.
Now, A A^-1 = I, A^-1 A = I, and [A I] row reduced exposes x (one way to find inverse), and there are a list of rules telling you if a matrix will be invertible (which you should review). If BA=I or AB=I, then B=A^-1 or A=B^-1. Also there is the topic of "a matrix composed of elementary matrices" which you should remember because you'll be needing it like a carpenter would a level.
So if your asked:
3x1+4x2=3
5x1+6x2=7
you have Ax=b (b={3,7}) and x=A^-1 b, you solve as you learned in earlier lessons
your new situation is no different: just re-worded
S(T(x))=S(Ax)=A^-1 (Ax)=x
"just ignore the new wording"
Back to your problem you have V, W, x they are all constant. I might say "find T for V->W so that Vx->W" but i wouldn't since you have all these defined as constant. That means I'm unsure of what the problems is asking.
"problem theory": ALWAYS ask if the problem is asking for a number as an answer or to change and equation from one form to another form as an answer. Do not try to solve a problem if the terms are unclear. You are given data but (as i am sitting) i have not been asked to do anything. So i should do nothing but demand i be told what it is I am being asked to do :) I could be getting asked for Wxb=V and what is b, but i cannot say i am. I could be getting asked WV=xC, do i know?
as far as invertible, i think i covered that above