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.
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.
