
\subsection{\bf The Definition of a Vector}

\begin{slide}

\begin{itemize}
\item Any vector $\underline{x} = \left( x_1, x_2, \ldots x_n
\right)$ can be written in terms of the complete basis
\[ \underline{x} = \sum_k x_k \underline{e}_k \]

\item $A_{ij}$ is representation of {\fifteenbxti A} in the basis $\lbrace
\underline{e}_k \rbrace$

\item The operator {\fifteenbxti A} operating on vector $\underline{x}$
is
\begin{eqnarray*}
\underline{y} & = & A \underline{x} \\
\underline{y} & = & \sum_l y_l\, \underline{e}_l \\
\underline{x} & = & \sum_k x_k\, \underline{e}_k \\
\hbox{Components are}\ y_l & = & \sum_k A_{lk}\, x_k
\end{eqnarray*}

\item Physicists often write
\[ \underline{x} = |x > \hbox{called a ket} \]
\end{itemize}

\end{slide}

