Definition
A linear system is a system in which inputs x_1 and x_2 produce the outputs y_1 and y_2 after being run through the system and being multiplied by complex constants, regardless of the order in which said operations are done.
Examples
-Linear System-
y(t) = x(t)
(1) 2x->syst.->y=2x->(*(2j+1))->z=4jy+2y=4jx+2x (2) 3x->syst.->y=3x->(*2)->z=6y=6x (1)+(2) 4jy+8y
Now using the opposite:
(1) 2x->(*(2j+1))->4jx+2x->syst.->z=4jy+2y=4jx+2x (2) 3x->(*2)->6x->syst.->z=6y=6x (1)+(2) 4jy+8y
They are the same=> The system is linear.
-Non-Linear System-
$ y(t) = x(t)^3 $
(1) 2x->syst.->$ y=8x^3 $->(*2)->z=2y=$ 16x^3 $ (2) 3x->syst.->$ y=27x^3 $->(*2)->z=2y=$ 54x^3 $ (1)+(2) $ 70x^3 $
Now using the opposite:
(1) 2x->(*2)->y=$ 4x $->syst.->$ z=y^3=64x^3 $ (2) 3x->(*2)->y=$ 6x $->syst.->$ z=y^3=216x^3 $ (1)+(2) $ 280x^3 $
The outcomes are not the same=> The system is not linear
