A system is linear if for specific inputs $ x_1(t) $ and $ x_2(t) $ yielding the outputs $ y_1(t) $ and $ y_2(t) $, respectively, the combination $ ax_1(t) + bx_2(t) $, where $ a $ and $ b $ are any complex numbers, yields the output $ ay_1(t) + by_2(t) $.
Example of a non-linear system:
$ System = \sqrt(t) $
$ x = 1 $, $ y = 2 $, $ a = 2 $, $ b = 1 $
$ x \Longrightarrow System \Longrightarrow 1 $
$ y \Longrightarrow System \Longrightarrow 1.41421 $
$ a(1) + b(1.41421) = 3.41421 $
$ ax + by \Longrightarrow System \Longrightarrow 2 \neq 3.41421 \therefore $ the system is not linear.
Example of a linear system:
$ System = t $
$ x = 1 $, $ y = 2 $, $ a = 2 $, $ b = 1 $
$ x \Longrightarrow System \Longrightarrow 1 $
$ y \Longrightarrow System \Longrightarrow 2 $
$ a(1) + b(2) = 4 $
$ ax + by \Longrightarrow System \Longrightarrow 4 \therefore $ the system is linear.
