In my textbook, it is stated that for a discrete system, where the input and output are expressed by difference equations, to be causal, there needs to be initial rest. It is also stated that for the system to be linear, the initial conditions should all be zero. I understand why these cases are true, however (I have found both conditions in most of the sites I searched) I do not understand why we need to talk about both of them. Is it because of the "linear" nature of the diff. equation that causality implies linearity, if that's the case?
My translation of what the book states: "We would also like that the difference equation corresponds to a causal system. A linear system is causal if the following condition holds: $$x(n) = 0 \forall n \leq n_0\implies y(n) = 0 \forall n \leq n_0$$ where $n_0$ is an arbitrary point in time, and so we say that the system is initially at rest.
Edit: I see now, that this condition is presented after it is stated that for the system to be linear, we need initial conditions to be zero. Would the condition for causality be subject to any change if we didn't care about linearity?