Linear System
Linear System is a system that is has the following rule:
If $ y_1(t) = H{x_1(t)} \, $
and
$ y_2(t) = H{x_2(t)} \, $
Thus,
$ ay_1(t) + by_2(t) = H(ax_1(t) + bx_2(t)) \, $
Example of a Linear system
$ x_1(t) = 2t \, $
$ x_2(t) = 4t \, $
$ y_1(t) = 2(x_1(t)) = 2(2t) \, $
$ y_2(t) = 2(x_2(t)) = 2(4t) \, $
$ ay_1(t) + by_2(t) = a2(2t) + b2(4t) \, $
$ = 2(a(2t) + b(4t)) \, $
$ = 2(ax_1(t) + bx_2(t)) \, $
It follows the linearity rule, therefore $ y(t) = 2(x(t)) \, $ a linear system.
Example of Non Linear System
$ x_1(t) = t^2 \, $
$ x_2(t) = t^3 \, $
$ y_1(t) = (x_1(t))^2 = (t^2)^2 = t^4\, $
$ y_2(t) = (x_2(t))^2 = (t^3)^2 =t^6\, $
$ ay_1(t) + by_2(t) = a(t^4) + b(t^6) \, $
$ \neq (a(t^2) + b(t^3))^2 \, $ which is equal to $ (a^2y_1(t) + b^2y_2(t)) \, $
As shown, it does not follow the rule, thus $ y(t) = (x(t))^2 \, $ not a linear system
