If I was given 3 equations, with an output of (x,y,z), if I were to project these outputs onto some basis: df/dx, df/dy, df/dz and a system exists that a1*df/dx + a2*df/dy + a3*df/dz + ... = (x,y,z), would I have achieved superposition?
Only if the system is linear, or linearizable. Also you have to be sure those basis vectors span the solution set of the system and have proper codomains (you want to make sure the projection is faithful).
