\documentclass[12pt,notitlepage]{article}
\usepackage{amsmath,amssymb,multicol,fullpage}
\usepackage[setpagesize=false]{hyperref}
\usepackage[all]{xy}
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}

\def\rank{{\operatorname{rank}}}
\def\bbF{{\mathbb F}}

\def\myurl{http://www.math.toronto.edu/~drorbn/}

\begin{document}
\noindent{\footnotesize
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1415}{2014-15}:
  \href{http://drorbn.net/?title=14-240}{Math 240}:
  November 12:
  \hfill\url{http://drorbn.net/?title=14-240}
}

\vskip 5mm
\centerline{\Large\bf Understanding}
\thispagestyle{empty}

\begin{multicols}{2}
\noindent $P$ onto, $Q$ 1--1 $\Longrightarrow$ $\rank(T) = \rank(PTQ)$:
\[ \xymatrix{
  V \ar[r]^T &
  W \ar@{^{(}->}[d]^P \\
  V' \ar@{->>}[u]^Q \ar[r]^{T'}_{PTQ} &
  W'
} \]

\noindent With $A=[T]_\beta^\gamma$, $\rank(A) = \rank(T)$:
\[ \xymatrix{
  V \ar[r]^T \ar[d]_{[\ ]_\beta} &
  W \ar[d]^{[\ ]_\gamma} \\
  \bbF^n \ar[r]^A_{T_A} &
  \bbF^m
} \]
\end{multicols}

\begin{multicols}{2}
\textbf{High school.} ``Understanding is a painful process. You only understand by doing many, many routine exercises.''

\textbf{Undergrad.} ``Understanding is a slow process. You only really understand a class once you take its continuation class.''

In both cases understanding is a passive thing. You do your thing and it comes when it wants.

What I learned only at the very end of my undergraduate degree is to turn ``understanding'' into a deliberate, active process. I still don't know how to explain it to others, yet if you think hard enough about a subject and truly try to decipher it, eventually you can - here and now, and not next year.

In fact, that process of ``understanding'' is not very different, at least in my mind, from the process called ``research''.
\end{multicols}

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%\noindent\rule{1in}{1pt}\newline
%{\footnotesize Created November 11, 2014, edited \today.}

\end{document}