Definition of Linearity
I don't like the diagrammed version of linearity much; I would much rather use the mathematical definition. As I mentioned here, I think of linearity like this:
The function ("The system") f is linear iff $ \forall x_1(t), x_2(t) \text{ and } \forall a,b \in \mathbb{C}, f(ax_1 + bx_2) = af(x_1) + bf(x_2) $
In layman's terms, that means that a system (call it f) is linear if functions (call them x and y) can be sent through the system in either one of these two ways and come out with the same result:
- Add some number a of x's to some number b of y's, then send them through the system together.
- Take x, send it through the system and then multiply the result by a. Take y, send it through the system, and multiply the result by b. Add the two together.
...for any and all choices of a, b, x, and y.
Example linear system
Let system f be defined for any function x as follows: f(x) = 2x. (In class, we would say "x --> system --> y = 2x".) We want to show f is linear.
Say we take any two functions $ x_1(t), x_2(t) $ and any two variables $ a,b \in \mathbb{C} $. (Try to pick them to prove me wrong!)
$ f(ax_1 + bx_2) = 2(ax_1 + bx_2) = 2ax_1 + 2bx_2 $
$ af(x_1) + bf(x_2) = a(2x_1) + b(2x_2) = 2ax_1 + 2bx_2 $
$ f(ax_1 + bx_2) = af(x_1) + bf(x_2) $
Since I was never explicit in my choice of $ x_1, x_2, a,\text{ or } b $ (I even let you choose them!), we can conclude that the above applies to every combination of $ x_1, x_2, a,\text{ and } b $ (within the constraints of the definition):
$ \forall x_1(t), x_2(t) \text{ and } \forall a,b \in \mathbb{C}, f(ax_1 + bx_2) = af(x_1) + bf(x_2) $.
Thus, f(x) = 2x is linear.
Example non-linear system
Let system f be defined for any function x as follows: f(x) = 2x-1. We want to show f is non-linear. (In class, we would say "x --> system --> y = 2x - 1".)
Say we take any two functions $ x_1(t), x_2(t) $ and any two variables $ a,b \in \mathbb{C} $. (This time, I'll choose them; we'll save that for later.)
$ f(ax_1 + bx_2) = 2(ax_1 + bx_2) - 1 = 2ax_1 + 2bx_2 - 1 $
$ af(x_1) + bf(x_2) = a(2x_1 - 1) + b(2x_2 - 1) = 2ax_1 + 2bx_2 - (a + b) $
Let me leave $ x_1 $ and $ x_2 $ alone; I don't care what they are. However, I would like for a = b = 1.
$ f(ax_1 + bx_2) = 2x_1 + 2x_2 - 1 $
$ af(x_1) + bf(x_2) = 2x_1 + 2x_2 - 2 $
$ f(ax_1 + bx_2) \neq af(x_1) + bf(x_2) $
Recall the definition for linearity:
$ \forall x_1(t), x_2(t) \text{ and } \forall a,b \in \mathbb{C}, f(ax_1 + bx_2) = af(x_1) + bf(x_2) $.
Since, in this case, the definition is not true for all functions and constants (The one above didn't work, for instance.), I can conclude that the system is not linear.
