Definition
A system is linear if:
Additive Property 1. The response to $ x_1(t) + x_2(t) $ is $ y_1(t) + y_2(t) $.
Scaling Property 2.The response to $ ax_1(t) $ is $ ay_1(t) $, where a is any complex constant.
Linear
$ y(t)=tx(t) $
Test the scaling property:
$ x_1(t)\rightarrow y_1(t)=tx_1(t) $ and $ x_2(t)\rightarrow y_2(t)=tx_2(t) $
$ x_3(t)=ax_1(t)+bx_2(t) $ a and b are arbitrary scalars
$ \begin{alignat}{4} y_3(t) & = tx_3(t) \\ & = t(ax_1(t)+ bx_2(t)) \\ & = atx_1(t)+ btx_2(t) \\ & = ay_1(t)+ by_2(t) \\ \end{alignat} $
So we can concluse that this system is linear
Non-Linear
$ y[n]=2x[n]+3 $
Test the additive property:
$ x_1[n]=2 $ and $ x_2[n]=3 $
$ y_1[n]=2x_1[n]+3=7 $
$ y_2[n]=2x_2[n]+3=9 $
$ y_3[n]=2x_3[n]+3=2(x_1[n]+x_2[n])+3=13 $
$ y_3[n]=13 \ne y_1[n]+y_2[n]=16 $
This shows that the additive property is violated, so the system is non-linear.
