Linear System Definition
A system takes a given input and produces an output. For the system to be linear it must preserve addition and multiplication. In mathematical terms:
$ x(t+t_0)=x(t) + x(t_0)\, $
and
$ x(kt)=kx(t)\, $
Linear System Example
Consider the system $ y[n]=x[n]\cdot\mathbf{M} $
let
$ \mathbf{a} = \begin{bmatrix}2 & 2 \end{bmatrix} $
$ \mathbf{b} = \begin{bmatrix}4 & 1 \end{bmatrix} $
$ \mathbf{M} = \begin{bmatrix}1 & 2 \\ 3 & 4 \\ \end{bmatrix} $
$ k=3\, $
If the system is linear these properties hold:
$ y[\mathbf{a}+\mathbf{b}]=y[\mathbf{a}]+y[\mathbf{b}] \, $
thus the system is linear.
$ y[k\mathbf{a}]=ky[\mathbf{a}] \, $
Here is the proof that the first prop holds:
$ y[a] = \begin{bmatrix}8 & 12 \end{bmatrix} $
$ y[a+b] = \begin{bmatrix} 6 &3 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} =\begin{bmatrix}24 & 18 \end{bmatrix} $
$ y[a]+y[b]= \begin{bmatrix}8 & 12 \end{bmatrix} +\begin{bmatrix}16 & 6 \end{bmatrix} = \begin{bmatrix}24 & 18 \end{bmatrix} $
And the second:
$ ky[\mathbf{a}] =\begin{bmatrix} 24 &36 \end{bmatrix} \, $
$ y[k\mathbf{a}] =\begin{bmatrix} 6 &6 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \, $ $ =\begin{bmatrix} 24 &36 \end{bmatrix} $
Non-Linear System Example
Consider the system $ y(t)=x(t)^2 \, $
let
$ x(t_0)=2\, $
$ x(t_1)=-2\, $
If the system is linear these properties hold:
$ y(t_0+t_1)=y(t_0)+y(t_1) \, $
$ y(kt)=ky(t) \, $
To see if the system is linear test the first property:
$ y(t_0)+y(t_1)= 4 +4 =8 \, $
$ y(t_0+t_1)= x(t_0+t_1)^2= 0\, $
Since $ 0\neq 8 $, the system isn't linear.
