% LESSON TWELVE
          
\documentclass{amsart}
\thispagestyle{empty}
\theoremstyle{definition}
\newtheorem*{dfn}{Definition}
\newtheorem*{notation}{Notation}
\begin{document}
             
\begin{center}
	\textbf{Lesson Twelve: The }\verb!cases! \textbf{ and } 
	\verb!array! \textbf{ Environments}
\end{center}  

\begin{dfn}
We define the \emph{absolute value function} $|x|$ as follows.
\end{dfn}

% We must be inside a math enviroment (here $$) to use the cases environment.
$$ |x|=
\begin{cases}
-x, &\text{if }x<0 \\
x, &\text{if }x\geq 0
\end{cases}
$$
% Note the use of | rather than \mid for the absolute value function.
% The latter results in unfelicitous spacing.

\begin{notation}
We denote the \emph{identity matrix} as $I_n$. See below.
\end{notation}

% We must be inside a math enviroment (here $$) to use the array environment.
$$ I_n=
\left ( 
% See Lesson Six for information on sizing delimiters as in \left (. 
\begin{array}{ccccc} 
% The c's center each column, as in the \tabular environment.
% See Lesson Ten for details.
1 & 0 & \cdots & 0 & 0 \\
0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & 0 \\
0 & 0 & \cdots & 0 & 1 \\
\end{array}
\right )
$$

% You might also consider using the \matrix environment, which
% is part of the amsmath package.

\end{document}