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\def\course{MATH 5112 (Spring 2024)}

\def\disclaimer{This document is a \textbf{transcript} of the lecture, so is more like an abbreviated set of lecture slides than complete lecture notes. For the latter, \textbf{consult the textbook} listed on the course webpage.}

\pagestyle{fancyplain}




%%% transcription begins here

\rfoot{}\cfoot{\thepage}\lfoot{}
\lhead{\text{\course}}\chead{}\rhead
{
 \text{Lecture ?? (transcribed by ??)}
}

\begin{document}


\disclaimer


\section{The First Section}
 
 Some text. Some \textbf{bold text}.
 
 \begin{theorem}\label{thm1} A theorem.
 \end{theorem}
 
 \begin{proof}
 Its proof.
 \end{proof}
 
  \begin{proposition} A proposition.
 \end{proposition}
 
 
  \begin{lemma} A lemma.
 \end{lemma}
 
 
  \begin{corollary} A corollary.
 \end{corollary}
 
 
   \begin{definition} A \defn{definition}. With labeled enumeration:
   \ben
   \item[(a)] item a
   \item[(b)] item b
   \een
 \end{definition}
 
  \begin{example} An example. With unlabeled itemization:
   \bei
   \item item a
   \item item b
   \eei
 \end{example}

    \begin{remark} A remark.
 \end{remark}
 
 
  \begin{notation} A remark about notation.
 \end{notation}

 
A reference to Theorem~\ref{thm1}.

A labeled displayed equation, referenced as \eqref{eq1}, showing scaled left/right brackets and cases:
\be\label{eq1}
A \eqdef \left\langle x_i : \prod_i x_i=1\right\rangle \quand
f(x) = \begin{cases} 
x+1 &	\text{if $x>0$}, \\
x^2+1 &	\text{otherwise}.
\end{cases}
\ee

An unlabeled, multi-line equation with matrices:
\[
\begin{aligned}
A &= \mat{{rrr} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 3 &2 & 1}, \\
v &= \vec{v_1 \\ v_2 \\ \vdots \\ v_n}.
\end{aligned}
\]

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  & \QSym  &  
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  \qquad
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\cG \arrow[r] \arrow[d] & \cO  \arrow[d]  \\ 
\cP \arrow[r] & \QSym   
  \end{tikzcd}
\]

 \end{document}