### Linearity

## Theory

There are three definitions we discussed in class for linearity.

__Definition 1__

__A system is called __**linear** if for any constants $ a,b\in $ *all complex numbers* and for any input signals *x*_{1}(*t*),*x*_{2}(*t*) with response *y*_{1}(*t*),*y*_{2}(*t*), respectively, the system's response to *ax*_{1}(*t*) + *b*x* _{2}(*t

*)*

*is*ay

*t*

_{1}(*) +*b

*y*

_{2}(

*t*).

__Definition 2__

If

$ x_1(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_1(t) $

$ x_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_2(t) $

then

$ ax_1(t) + bx_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow ay_1(t) + by_2(t) $

for any $ a,b\in $ *all complex numbers*, any *x*_{1}(*t*),*x*_{2}(*t*) then we say the system is **linear**.

__Definition 3__

## Applications

Linearity can be used to simplify the Fourier transform. Integration and differentiation are also linear. Once a non-linear system is made linear, complex systems are easier to model mathematically. True linear systems are virtually unknown in the real world, but over a small range of variables, systems can be modeled as linear.