Definition
If Z(t) and W(t) in the following are equal the system is linear.
Examples
Linear:
$ \ S_{1} = 2x(t + 3) + x(t - 8) $
$ \ S_{2} = x(t - t_{0}) $
$ \ S_{1} \rightarrow S_{2} = 2x(t - t_{0} + 3) + x(t - t_{0} - 8) $
$ \ S_{2} \rightarrow S_{1} = 2x(t - t_{0} + 3) + x(t - t_{0} - 8) $
Since the results are the same the system is time invariable.
Non-Linear:
$ \ S_{1} = x(-t + 3) - x(-t - 8) $
$ \ S_{2} = x(t - t_{0}) $
$ \ S_{1} \rightarrow S_{2} = x(-t + t_{0} + 3) - x(-t + t_{0} - 8) $
$ \ S_{2} \rightarrow S_{1} = x(-t - t_{0} + 3) - x(-t - t_{0} - 8) $
Since the results are different they system is time variant.
