here is a picture of my problem 
Basically what i have is that i was told i could find this homomorphism by doing the following
assume
u=f(x,y,z)
v=g(x,y,z)
w=h(x,y,z)
then
u'=(fx)x' +(fy)y' +(fz)z'
v'=(gx)x' +(gy)y' +(gz)z'
w'=(hx)x' +(hy)y' +(hz)z'
From this apparently i reconstruct what f,g,and h have to be. I suppose my question regards what is the point of doing this method? how do i progress here? x',y',z' are known however u' v' and w' are unknown and the im not sure how to reconstruct the functions to gain this idea of a homomorphism between the linear and nonlinear system.
In addition to attempting this, i have actually found analytical solutions to this system where
x(t)=e^(-2t)
y(t)=-(e^(-6t))/7
z(t)=(e^(-8t))/35
I dont know if that is specifically helpful to finding the homomorphism or not.
Some insight would be helpful. Thank you.