\documentclass[11pt]{amsart}
\usepackage{amssymb,amsmath}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}

\begin{document}

Here is a generally looking matrix: 

$$ \left( \begin{array}{cccc} 
            a_{11} & a_{12} & \ldots & a_{1n} \\
            a_{21} & \ldots &        &        \\
            \vdots &        &        &        \\
            a_{m1} & \ldots &        & a_{mn}   
                \end{array} \right) $$    

If $X$ is a matrix with entries in $\C$, say  

\begin{equation}  
     X = \left( \begin{array}{cc} 
                   \alpha &   \beta \\
                    \gamma &  \delta    \end{array} \right). 
\end{equation} 

the complex conjugate transpose $X^*$ is easy to write: 

\begin{equation}  
     X^* = \left( \begin{array}{cc} 
                   \bar{\alpha} &   \bar{\gamma} \\
                    \bar{\beta} &  \bar{\delta}    \end{array} \right). 
\end{equation} 


Solve the following problem and write your solution in \LaTeX :

Prove that the set of all $2 \times 2$ 
complex matrices $X$ such that $X^* +X =0$ and ${\rm tr}(X)=0$, is a linear 
space over $\R$.   

Find the dimension and a basis of this space. 


\end{document}

Use the theorem and proof environments to state your result. 
Use the title environment (see Sample 3) to sign and date your solution. 
