Linearity
A system is called linear if it's output can be shown as a sum of it's inputs prior to being input to the system.
Example of a Linear System
Given the system $ y(t) = 5x(t) $
If we apply two input signals $ x_1(t) = 4n $ and $ x_2(t) = 5 $, we end up with two outputs $ y_1(t) = 20n $ and $ y_2(t) = 25 $.
If we apply the sum of the two inputs $ x_3(t) = 4n + 5 $, we end up with the output $ y_3(t) = 20n + 25 $.
Since $ y_1(t) + y_2(t) = y_3(t) $ the system is linear.
Example of a Non-Linear System
Given the system $ y[n] = x[n]^{2} $
If we apply two input signals $ x_1[n] = 4x $ and $ x_2[n] = 5 $, we end up with two outputs $ y_1[n] = 4x^2 $ and $ y_2[n] = 25 $.
If we apply the sum of the two inputs $ x_3[n] = 4x + 5 $, we end up with the output $ y_3[n] = 16x^2 + 40x + 25 $.
Since $ y_1[n] + y_2[n] \ne y_3[n] $ the system is NOT linear.
