A linear system is one that, regardless of the steps taken to return an output, will return the same output. The input to the system in each case must be the same. Also, steps to return an output only include addition of signals, multiplication by complex numbers, and the trip through the system itself. A possible analogy could refer to a "black box" of sorts. For example, two black boxes have the same specifications, but the implementation is different. Both boxes return the same output given the same set of inputs.
Example: A system that is linear
$ system -> y(t)=x(4t) $
$ x_1(t), x_2(t), a=\frac{1}{3}, b=\frac{1}{6} $
Multiply by a & b, add, then put through system.
$ x_1(t)*a = ax_1(t), x_2(t)*b = bx_2(t) $
$ y_1(t)=ax_1(4t)+bx_2(4t) $
Put through system, multiply, then add.
$ x_1(t) \rightarrow x_1(4t), x_2(t) \rightarrow x_2(4t) $
$ x_1(4t) \rightarrow ax_1(4t), x_2(4t) \rightarrow bx_2(4t) $
$ y_2(t)=ax_1(4t)+bx_2(4t) $
$ y_1 $ and $ y_2 $ are equal; therefore, system is linear.
Example: A system that is not linear
$ system -> y(t)=x(t)^3 $
$ x_1(t), x_2(t), a=\frac{1}{3}, b=\frac{1}{6} $
Multiply by a & b, add, then put through system.
$ x_1(t)*a = ax_1(t), x_2(t)*b = bx_2(t) $
$ y_1(t)=(ax_1(t)+bx_2(t))^3 $
Put through system, multiply, then add.
$ x_1(t) \rightarrow x_1(t)^3, x_2(t) \rightarrow x_2(t)^3 $
$ x_1(t) \rightarrow ax_1(t)^3, x_2(t) \rightarrow bx_2(t)^3 $
$ y_2(t)=ax_1(t)^3+bx_2(t)^3 $
$ y_1 $ and $ y_2 $ are not equal; therefore, system is not linear.
