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[a+b]=y[a]+y[b] \, $
$ y[ka]=ky[a] \, $
Here is the proof that the first prop holds:
$ y[a] = \begin{bmatrix}8 & 12 \end{bmatrix} $
$ y[a+b] = \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[a] = \, $
