My Definition
A system is call linear if the system is "distributive".
system(input1+input2)=system(input1)+system(input2)
Example of Linear System
The system is y(t)=2x(t+3).
The output of the signal x1(t)=t is y1(t)=2(t+3)=2t+6.
The output of the signal x2(t)=t-1 is y2(t)=2(t+3-1)=2t+4.
What is the output of the signal x(t)=x1(t)+x2(t)=t+t-1?
y(t)=2(2(t+3)-1)=2(2t+5)=4t+10
This system is linear beacause
y(t)=4t+10
y1(t)+y2(t)=2t+6+2t+4=4t+10
Example of Non-Linear System
$ y(t)=(x(t))^2 $
$ x_1(t)=t $ yields $ y_1(t)=t^2 $
$ x_2(t)=t-1 $ yields $ y_2(t)=(t-1)^2=t^2-2t+1 $
But,
$ (x(t)=t+t-1 $ yields $ y(t)=(t+t-1)^2=(2t-1)^2=4t^2-4t-1 $
and
$ y(t)=y_1(t)+y_2(t)=t^2+t^2-2t+1=2t^2-2t+1 $
which are clearly not the same.
