Time Invariance
A time-invariant system is a system in which the output gets time-shifted when the input is time-shifted.
$ x(t - t_0) \rightarrow system \rightarrow y(t - t_0) $
Time-invariant System
An example of a time-invariant system would be the system I used for my linearity problem. Therefore the system is a linear, time-invariant system.
$ x(t) \rightarrow system \rightarrow y(t) = 2x(t) $
Proof:
$ x(t) \rightarrow system \rightarrow 2x(t) \rightarrow time-delay \rightarrow 2x(t-t_0) $
$ x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2x(t-t_0) $
Since the output is the same for both configurations the system is time-invariant.
Time-variant System
An example for a time-variant system would be $ x(t) \rightarrow 2tx(t) $
Proof:
$ x(t) \rightarrow system \rightarrow 2tx(t) \rightarrow time-delay \rightarrow 2(t-t_0)x(t-t_0) $
$ x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2tx(t-t_0) $
