A function \(f\) is called a linear operator if it has the two properties:
- \(f(x+y)=f(x)+f(y)\) for all \(x\) and \(y\);
- \(f(cx)=cf(x)\) 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”.
