\part{Mathematical foundations}

Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots

\chapter{Linear algebra}

\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 \vec{e}_i a_i
\label{eq:vec_sum}
\end{equation}
The scalar product for an $N$-dimensional vector space is defined as
\begin{equation}
(\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
\end{equation}
which introduces a norm
\begin{equation}
||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
\end{equation}
that correpsonds to the length of vector \vec{a}.
Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} \ldots
\begin{equation}
\end{equation}

