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 \myhead{Mat 5320}{Final Exam}{Autumn '05}{}
 
 You must work alone. You may use the text, other books, your notes, etc.~and your instructor; you may not consult another person in any way. Answer all questions completely. \\[1ex]
 
 %%%%%%%%
\S I. \textsc{True} and/or \textsc{False}. There are \ref{LastTF} questions at 2 points each. Circle your answer.
\begin{enumerate}
\item \TF{For any finite $n\in\mathbb{Z}^{+}\!\!,$ there is only one real vector space of dimension $n.$ } \\
	
\item \label{LastTF} 
	\TF{Each $m\times n$ matrix with elements in $\mathbb{Z}_5$ defines a unique linear transformation.} \\[3ex]
\end{enumerate}

 
 %%%%%%%%%
 \S II. \textsc{Problems}. There are \ref{LastProb} problems at $20$ points each.
 
\begin{enumerate}
\item Let $A \in L(\mathbb{R}^4,\mathbb{R}^4)$ be given by $A = \begin{bmatrix} 0&-1&0&2 \\ -1&0&0&2 \\ 0&0&1&0 \\ -1&1&0&1 \end{bmatrix}\!.$ Find
\begin{enumerate}
\item $p(\lambda)$
\item $m(\lambda)$
\item the eigenvalues and their algebraic \& geometric multiplicities
\item the transition matrix $P$
\item $JCF(A)$ \\
\end{enumerate}

\item Determine all possible Jordan Canonical Forms for a $5\times 5$ matrix $A$ 
	\begin{enumerate}
	\item whose characteristic polynomial is $p(\lambda)=(2-\lambda)^3(5-\lambda)^2.$
	\item whose minimal polynomial is $m(\lambda)=(4-\lambda)^2.$ \\
	\end{enumerate}


\item Prove:
	 \begin{theorem} Define the $k\times k$ matrix 
	 \[  N_k= \begin{bmatrix}
		0 & 1 & 0 & \dots & 0 \\
		0 & 0 & 1 & \dots & 0 \\
		\vdots &  & \ddots & \ddots & \vdots \\
		0 & &  \dots & 0 & 1 \\
		0 & & \dots & 0 & 0 \end{bmatrix}. \]
	Then $N_k$ is nilpotent of index $k.$ \\
	 \end{theorem}


\item \label{LastProb} Prove:
	 \begin{theorem} Let $A\in \mathfrak{M}_{n\times n}.$ $A$ is nonsingular iff the characteristic polynomial has nonzero constant term; i.e., $p(0)\ne 0.$  \\[4ex]
	 \end{theorem}

\end{enumerate}


\textsc{ec}: Who said: "You can observe a lot just by watchin'.''
 
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