Subsection 2.1.5 Linear Operators
An entity \(\hat{A}\) which relates every vector \(\psi_{i}\) in a vector space \(V_{n}\) to another vector \(\phi_{i}\) in this space by the equation
\begin{equation}
\phi_{i}=\hat{A} \psi_{i}\tag{2.1.9}
\end{equation}
is called an operator. The operator \(\hat{A}\) is said to be linear if it possesses the following properties
\begin{equation}
\hat{A} (\psi_{a}+\psi_{b}) = \hat{A} \psi_{a}+\hat{A} \psi_{b})\tag{2.1.10}
\end{equation}
\begin{equation}
\text{and}\quad \hat{A} (\lambda \psi) = \lambda\hat{A} \psi\tag{2.1.11}
\end{equation}
where \(\lambda\) is a scalar (a real or complex number).
