Linear system
A linear system is a system that will produce the same output for both of the following actions:
1. One puts signals through the system, multiplies the outcomes by a constant, and add the results together.
2. One multiplies the same signals by the same constants, adds the results together, and sends that outcome through the system.
An example of a linear system is as follows:
$ y(t) = 15x(t)\! $
The proof for this is rather simple. Suppose you put $ x(t) = t + 12\! $ and end up with $ 15t + 180\! $. You also send $ z(t) = t - 2\pi\! $ through the system and get $ 15t - 30\pi\! $. You multiply the first outcome by $ 2\! $ and get $ 30t + 360\! $. You multiply the second result by $ 3\! $ and you get $ 45t - 90\pi\! $. After summing the two, you get $ 75t + 360 - 90\pi\! $.
Now, we do it the second way. First, we multiply the signals by the given constants. $ x(t) = t + 12\! $ becomes $ 2t + 24\! $, and $ z(t) = t - 2\pi\! $ becomes $ 3t - 6\pi\! $. Then, we add them together, and we get $ 5t + 24 - 6\pi\! $. Finally, we run it through the system, which gives us $ 75t + 360 - 90\pi\! $. As you can see, it checks.
Non-linear system
An example of a non-linear system is as follows:
$ y(t) = [x(t)]^2\! $
Suppose you put $ x(t) = 12t\! $ and end up with $ 144t^2\! $. You also send $ z(t) = 5t\! $ through the system and get $ 25t^2\pi\! $. You multiply the first outcome by $ 2\! $ and get $ 288t^2\! $. You multiply the second result by $ 3\! $ and you get $ 75t^2\pi\! $. After summing the two, you get $ 363t^2\! $.
Now, we do it the second way. First, we multiply the signals by the given constants. $ x(t) = 12t\! $ becomes $ 24t\! $, and $ z(t) = 5t\pi\! $ becomes $ 15t\! $. Then, we add them together, and we get $ 39t\! $. Finally, we run it through the system, which gives us $ 1521t^2\! $. As you can see, it does not checks at all. Thus, this system is non-linear.
