Definition
A system is linear if, when a weighted sum of individual inputs is presented to the system, the output consists of a weighted sum of the outputs due to the individual inputs.
A Linear System
Take, for instance, the system $ y(t)=17x(t-1) $.
When we apply an input signal $ x(t)=t^2 $, the output is $ y(t)=17(t^2-2t+1) $.
When we apply a different input signal $ x(t)=4t $, the output is $ y(t)=68(t-1) $.
When we apply the sum of the two input signals into our system, that is, $ x(t)=t^2+4t $, we see the following response:
$ y(t)=17[(t^2-2t+1)+4(t-1)] $
$ y(t)=17(t^2-2t+1+4t-4) $
$ y(t)=17(t^2+2t-3) $
the last line of which is the sum of our two original outputs. Since the system produces an output corresponding to the sum of the two input signals, the system is linear.
A Non-Linear System
An example of a non-linear system is $ y(t)=e^{x(t)} $.
When we apply an input signal $ x(t)=t^2 $, the output we see is $ y(t)=e^{t^2} $.
When we apply a second input signal $ x(t)=4t $, we achieve the response $ y(t)=e^{4t} $.
When we apply the sum of the two input signals, or $ x(t)=t^4+4t $, we see the following output:
$ y(t)=e^{t^2+4t} $
$ y(t)=e^{t^2}e^{4t} $
which is not the sum of the two original responses; rather, it is the product of the two original inputs. This goes against the definition of linearity, so the system is not linear.
