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\begin{document}

\begin{center}
\large{Math 262}\\
\large{Spring 1999}\\

\large{Test 1}
\end{center}

\bigskip

\begin{enumerate}

\item Write an essay about $F [X]$, the ring of polynomials with
coefficients in the field $F$.

\bigskip

\item Prove that a real $2 \times 2$ symmetric matrix is
diagonalizable.

\bigskip

\item Let $N$ be a $2 \times 2$ complex matrix, $N^2 = 0$.  Prove
that either $N = 0$ or $N$ is similar to 
$\left[
\begin{matrix}
0 &0 \\
1 &0
\end{matrix}
\right] \cdot$

\bigskip

Hint:  If $N \neq 0$ then $dim \ ker \ N = 1$ (why?).  Look at the
effect of $N$ on a basis $\{\alpha, \beta \}$ with $\beta \ \epsilon
\ ker
\ N$.

\bigskip

\item Any $2 \times 2$ complex matrix $A$ is either
diagonalizable or $A$ is similar to a matrix of the form 
$\left[
\begin{matrix}
a &0 \\
1 &a
\end{matrix}
\right]$

\bigskip

Hint:  In case the eigenvalues of $A$ are equal to $c$, show that
the eigenvalues of $N = A -cI$ are all zero.  Conclude that $N$ is
nilpotent and use problem 3. 

\end{enumerate}
\end{document}