Linearity
Linear Systems are systems that can be broken into parts where each part is solved independently and finally the solution can be made by the recombination (summation) of these parts.By contrast Non-Linear systems generally cannot be split up in this way. The different components of the system depend on each other. Therefore one cannot reduce the problem to a simpler statement of the problem and usually due to combinatorial-explosion the solution space is massive and hence difficult to solve.
For example: listening to your two favourite songs at the same time doubles the noise you hear (Linearity) but it doesn't double the pleasure (Non-Linearity).
This can be expressed Mathematically as a linear map or linear function,
f(x) is a function that satisfies the following two properties: [1] Additivity : f(x + y) = f(x) + f(y). [2] Homogeneity: f(αx) = αf(x) for all α, (Homogeneity of degree 1)
- Equation [1] just says that a function of the two variables x and y can be split and summed into two functions. Just what we said above.
- Equation [2] is a bit cooler still. It says that we can really decompose x itself. So for example, 50 = 50 x 1,
so f(50) = 50f(1). Normally calculating any function on 1 is much easier that doing it on some arbitrary number like 50.
| 07dec16 | admin |