1 \part{Mathematical foundations}
3 Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
5 \chapter{Linear algebra}
7 \section{Vectors and bases}
9 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 ${\vec{e}_i}$.
11 \vec{a}=\sum_i \vec{e}_i a_i
14 The scalar product for an $N$-dimensional vector space is defined as
16 (\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
18 which introduces a norm
20 ||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
22 that correpsonds to the length of vector \vec{a}.
23 Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} \ldots