# Linear operator

A function $f$ is called a linear operator if it has the two properties:

1. $f(x+y)=f(x)+f(y)$ for all $x$ and $y$;
2. $f(cx)=cf(x)$ for all $x$ and all constants $c$.

It follows that $f(ax+by)=af(x)+bf(y)$ for all $x$ and $y$ and all constants $a$ and $b$.

The most common examples of linear operators met during school mathematics are differentiation and integration, where the above rule looks like this: $\begin{gather*} \frac{d}{dx}(au+bv)=a\frac{du}{dx}+b\frac{dv}{dx}\\ \int_r^s (au+bv)\,dx=a\int_r^s u\,dx+b\int_r^s v\,dx, \end{gather*}$

where $u$ and $v$ are functions of $x$, $a$ and $b$ are constants, and $r$ and $s$ are the limits of integration.

See also linear function for a subtly different use of the term “linear”.