A **linear system** must simultaneously verify the properties of **additivity** and **homogeneity**.
**Additivity** property:
$
\begin{array}{c}
S(x_1(n)) = y_1(n)\\
S(x_2(n)) = y_2(n)\\
\end{array}
\xrightarrow[additivity]{} S(x_1(n) + x_2(n)) = y_1(n) + y_2(n)
$
**Homogeneity** property:
$S(x(n)) = y(n) \xrightarrow[homogeneity]{} S(a x(n)) = a y(n), a \in \mathbb{R}$
Where $a$ is an arbitrary constant.
Examples of linear systems:
- $y(t) = tx(t)$
- $y(n) = x^2(n)$
Examples of nonlinear systems:
- $y(n) = \Re(x(n))$
- $y(t) = 2x(t) + 3$