Definition of Linearity
A system is linear if for any inputs $ \,x_1(t), x_2(t)\, $ yielding outputs $ \,y_1(t), y_2(t)\, $, respectively, the response to
$ \,ax_1(t)+bx_2(t)\, $ is
$ \,ay_1(t)+by_2(t)\, $, where $ \,a,b\in \mathbb{C}, a\not= 0 ,b\not= 0\, $.
Example of a Linear System
The following system is linear:
$ \,s(t)=2x(t+3)\, $
Proof:
We have two functions: $ \,x_1(t), x_2(t)\, $.
After applying the functions to the system $ \,s(t)\, $, we get:
$ \,y_1(t)=2x_1(t+3)\, $
$ \,y_2(t)=2x_2(t+3)\, $
Thus,
$ \,ay_1(t)+by_2(t)=\, $
$ \,a(2x_1(t+3))+b(2x_2(t+3))=\, $
$ \,2ax_1(t+3)+2bx_2(t+3)\, $
Now, apply $ \,ax_1(t)+bx_2(t)\, $ to the system $ \,s(t)\, $:
$ \,2(ax_1(t+3)+bx_2(t+3))=\, $
$ \,2ax_1(t+3)+2bx_2(t+3)\, $
Since the two results are equal
$ \,2ax_1(t+3)+2bx_2(t+3)=2ax_1(t+3)+2bx_2(t+3)\, $
the system is linear.
Example of a Non-Linear System
The following system is non-linear:
$ \,s(t)=2x(t)+3\, $
Proof:
We have two functions: $ \,x_1(t), x_2(t)\, $.
After applying the functions to the system $ \,s(t)\, $, we get:
$ \,y_1(t)=2x_1(t)+3\, $
$ \,y_2(t)=2x_2(t)+3\, $
Thus,
$ \,ay_1(t)+by_2(t)=\, $
$ \,a(2x_1(t)+3)+b(2x_2(t)+3)=\, $
$ \,2ax_1(t)+3a+2bx_2(t)+3b\, $
Now, apply $ \,ax_1(t)+bx_2(t)\, $ to the system $ \,s(t)\, $:
$ \,2(ax_1(t)+bx_2(t))+3=\, $
$ \,2ax_1(t)+2bx_2(t)+3\, $
Since the two results are not equal
$ \,2ax_1(t)+3a+2bx_2(t)+3b\not= 2ax_1(t)+2bx_2(t)+3\, $
the system is non-linear.
