more math

[lectures/latex.git] / physics_compact / math.tex

diff --git a/physics_compact/math.tex b/physics_compact/math.tex
index 0e49a40..5f437fa 100644 (file)
--- a/physics_compact/math.tex
+++ b/physics_compact/math.tex
@@ -1,6 +1,26 @@
 \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 ${\vec{e}_i}$.
+\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}