A linear function defined over a vectorial space (The set of real numbers with the usual operation of addition and multiplication form a vectorial space) has to satisfy the relations:


\begin{align}f(\lambda x) = & \lambda f(x) \notag \\
f(x+y) = & f(x)+f(y) \notag
\end{align}
where $x$ and $y$ are elements of the vectorial space and $\lambda$ is a scalar (a real or a complex number).

A linear operator $D$ has to satisfy the relations


\begin{align}D (\lambda f) & = \lambda D f \notag \\
D (f+g) & = D f+ D g \notag
\end{align}
where $\lambda$ is a scalar and $f$ and $g$ are elements of a functional vector space. An example of a linear operator is the Laplace's equation.


Timothy S Choe
2003-03-15