% Homework template for MA 614, Spring 2011.  When a line begins with the percent sign, the typesetter ignores it.  So, use percent signs at the beginning of lines to insert comments to yourself.


% Set the document class.  The command [11pt] sets the font at 11 point, which is nicer to read.  The default would be 10pt
\documentclass[11pt]{amsart} 


% Call packages that allow you to invoke certain mathematical symbols.
\usepackage{amssymb,amsmath,amsthm}


% Set the title, author, and date information.
\title{Homework Template}
\author{The Author}
\date{Monday, Jan 10, 2011.}


% Formally begin the document and make the title.
\begin{document}
\maketitle

This is the set of solutions to homework number 1.

\section{Problem 1 Solution}


% When you want to write mathematical symbols, put it between dollar signs.  If you want it to be displayed, put it between the symbols \[ and \].
If we let $f\colon X\rightarrow Y$ be the function we are considering, then we may study it.
Therefore, the answer is what we want it to be, which is the function 
\[g\colon \mathbb{Z}\rightarrow \mathbb{R} \, ,\] 
satisfying $g(0)=89$.

Note that in the latex file, I put a slash comma before my comma in the displayed equation, which adds a space and makes it look nicer.

Here are some various things you can write in Latex.  Thanks to Prof. Kate Ponto for making part of this file.  See the latex code to find out what to type to make these symbols appear.

$5^2$, $H_*$, $\frac{3}{2}$, $\int_0^1$
\[\int_0^1 \] $\sum_0^1$
\[\sum_{n=2}^{400}\]

If you want to make a numbered list, type this.

\begin{enumerate}
\item first choice %comment
$$ 4*3$$
\item second choice
\item third choice
\end{enumerate}

To make a bulleted list, type this.

\begin{itemize}
\item first choice
\item second choice
\item third choice
\end{itemize}

\section{Problem 2: Matrices and equations}

For matrices, you must type in the following.  See the AMS file linked to from the course website for more details on how to make matrices.

\[ \left[
\begin{array}{ccc}
2&3&4\\
4&5&6
\end{array}
\right]\]

To make a series of equalities that are lined up properly, you can again use the ``array'' environment.

$\begin{array}{rcl}
B-\lambda I&=& PAP^{-1}-\lambda I\\
&=&PAP^{-1}-\lambda PP^{-1}\\
&=& P(A-\lambda I)P^{-1}
\end{array}$

Use it again to define a function piecewise.

$f(x)=\left\lbrace\begin{array}{ll}0&\mathrm{if}\,\,x\in \mathbb{Q}\\
1&\mathrm{if}\,\,x\in\mathbb{R}\setminus \mathbb{Q}
\end{array}
\right.$

To make extra vertical space if you want to draw a picture on your printout, use

\vspace{2cm}

% This put 2cm of vertical space.  You can also use mm or in for milimeters or inches, respectively.

If you like to put a special symbol at the end of your proofs, you can try:

$\Box$
$\Diamond$

\end{document}