\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 ${\vec{e}_i}$
\label{eq:vec_sum}
\end{equation}
i.e., if the basis set is complete, any vector can be written as a linear combination of these basis vectors.
-The scalar product for an $N$-dimensional real vector space is defined as
+The scalar product in an $N$-dimensional Euclidean 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
+which satisfies the properties of an inner product (see \ref{math_app:product}) and 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}.
+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)=