I will use uppercase letters to denote known variables and lowercase letters to denote unkown ones.
Assume I have a vector $<X', Y', z'>$, and an invertible matrix $T$, and a vector $<x,y,z>$ constrained to a known plane $P=(N,V)$ ($N$ being the normal and $V$ being a point in the plane).
We know that $<X', Y', z'> = T\times<x,y,z>$.
Is it possible to either find $z'$ or $<x,y,z>$ in this system?
From the fact that $r = (x,y,z)$ lies in the plane you get: $$N^T(r-V)= 0$$ From the other equation you have: $$q = Tr$$ Where $q = (X', Y', z')$. Note that $T$ is invertible so the equations are not linearly dependent. Then ultimately you get 4 equations for 4 unknowns.