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 \part{Mathematical foundations}
 
+Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
+
 \chapter{Linear algebra}
 
-Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
+\section{Vectors and bases}
+
+A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_space} for mathematical details) is represented by its components $a_i$ with respect to a set of $N$ basis vectors ${\vec{e}_i}$.
+\begin{equation}
+\vec{a}=\sum_i^N \vec{e}_i a_i
+\label{eq:vec_sum}
+\end{equation}
+The scalar product for an $N$-dimensional real vector space is defined as
+\begin{equation}
+(\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
+\label{eq:vec_sp}
+\end{equation}
+which enables to define a norm
+\begin{equation}
+||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
+\end{equation}
+that just corresponds to the length of vector \vec{a}.
+Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} leads to
+\begin{equation}
+(\vec{a},\vec{b})=(\sum_i\vec{e}_ia_i,\sum_j\vec{e}_jb_j)=
+\sum_i\sum_j(\vec{e}_i,\vec{e}_j)a_ib_j \text { ,}
+\end{equation}
+which is equal to \eqref{eq:vec_sp} only if
+\begin{equation}
+(\vec{e}_i,\vec{e}_j)=
+\delta_{ij}  = \left\{ \begin{array}{lll}
+0 & {\rm for} ~i \neq j \\
+1 & {\rm for} ~i = j   \end{array} \right.
+\text{ (Kronecker delta symbol),}
+\end{equation}
+i.e.\  the basis vectors are mutually perpendicular (orthogonal) and  have unit length (normalized).
+Such a basis set is called orthonormal.
+The component of a vector can be obtained by taking the scalar product with the respective basis vector.
+\begin{equation}
+(\vec{e}_j,\vec{a})=(\vec{e}_j,\sum_i \vec{e}_ia_i)=
+\sum_i (\vec{e}_j,\vec{e}_i)a_i=
+\sum_i\delta_{ij}a_i=a_j
+\end{equation}
+Inserting the expression for the coefficients into \eqref{eq:vec_sum}, the vector can be written as
+\begin{equation}
+\label{eq:complete}
+\vec{a}=\sum_i \vec{e}_i (\vec{e}_i,\vec{a}) \Leftrightarrow
+\sum_i\vec{e}_i\cdot \vec{e}_i=\vec{1}
+\end{equation}
+if the basis is complete.
+Indeed, the very important identity representation by the outer product ($\cdot$) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.