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diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index 1903ab1..5f437fa 100644
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -1,6 +1,26 @@
-\chapter{Mathematical tools}
+\part{Mathematical foundations}
 
-\section{Spherical coordinates}
+Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
 
-\section{Fourier integrals}
+\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}