+Indeed, the very important identity representation by the outer product ($\otimes$, see \ref{math_app:product}) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.
+
+\section{Operators, matrices and determinants}
+
+An operator $O$ acts on a vector resulting in another vector
+\begin{equation}
+O\vec{a}=\vec{b} \text{ ,}
+\end{equation}
+which is linear if
+\begin{equation}
+O(\lambda\vec{a}+\mu\vec{b})=\lambda O\vec{a} + \mu O\vec{b} \text{ .}
+\end{equation}
+Thus, for a linear operator, it is sufficient to describe the effect on the complete set of basis vectors, which enables to describe the effect of the operator on any vector.
+Since the result of an operator acting on a basis vector is a vector itself, it can be expressed by a linear combination of the basis vectors
+\begin{equation}
+O\vec{e}_i=\vec{e}_jO_{ji}
+\text{ ,}
+\end{equation}
+with $O_{ji}$ determining the components of the new vector $O\vec{e}_i$ along $\vec{e}_j$.
+
+\section{Dirac notation}