\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^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.

