Linearity is defined as a system that contains superposition in the book(Signals and Systems 2nd ed. Oppenheim, 53). How I see it is if the input signal has a magnitude applied to it the output should have a magnitude applied to it. Also if two signals are added it would be as if each signal had went through the system and then had been added.
Linear System Example
$ Y(t) = x(t) $
$ ax_1(t)+bx_2(t) -> [System] -> aY_1(t) + bY_2(t) $
$ a_1(t)+x_2(t) -> [System] -> y_1(t) + Y_2(t) -> [a b] -> aY_1(t) + bY_2(t) $
The outcome of people ways are equal so it is Linear.
Non-Linear System Example
$ Y(t) = x(t)^2 $
$ x_1(t),x_2(t) -> [System] -> Y_1(t),Y_2(t) -> [a,b] -> aY_1(t),bY_2(t) -> add -> ax_1(t)^2 + bx_2(t)^2 $
$ x_1(t),x_2(t) -> [a,b] -> ax_1(t),bx_2(t) -> add -> ax_1(t) + bx_2(t) -> [System] -> [ax_1(t) + bx_2(t)]^2 $
They are not equal so it is not linear.
