\documentstyle[12pt]{article}
\input amssym.def
\input amssym
%\pagestyle{empty}

\newcount\probno
\probno=0
\def\prob{\advance\probno by 1 \bigskip\par\noindent\the\probno . }
\def\entry{{\rm entry}}
\def\span{{\rm span}}
\def\mat{{\rm MAT}}
\def\col{{\rm col}}
\def\reals{{\bf R}}
\def\range{{\rm range}}
\def\nullsp{{\rm null}}
\def\complex{{\Bbb C}}

\begin{document}

\centerline{A. Miller \hfill M340 \hfill Final \hfill May 97}

\bigskip

You may use without proof any theorem proved in class;
however when you do so, you should
{\bf state the result} you are using.
You may also use any
exercise, but in this case you should {\bf state the exercise and
give a full proof} of the exercise in your solution.

\bigskip

\prob Let $A$ be an $m\times n$ matrix, $B$ an $n\times p$ matrix
and $a$ a scalar. Prove that
$$a(AB)=(aA)B=A(aB).$$

\prob Prove that

\centerline{$v_1,\ldots,v_n$ are linearly dependent}

\centerline{iff}

\centerline{$v_1=z$ or $v_j\in \span(\{v_1,\ldots,v_{j-1}\})$ for some $j$ with
$1<j\leq n$.}



\prob
\begin{enumerate}
\item For $L:V\to W$ a function from the vector space $V$ to the
  vector space $W$, define $L$ is a linear transformation.
\item For $L:V\to W$ a linear transformation, define $\nullsp(L)$.
\item For $L:V\to W$ a linear transformation, define $\range(L)$.
\item For $L:V\to W$ a linear transformation, define $L$ is one-to-one.
\item For $L:V\to W$ a linear transformation, define $L$ is onto.
\item For $\lambda\in\complex$ and $A$ an $n\times n$ matrix, define
          $\lambda$ is an eigenvalue of $A$.
\item For $\lambda\in\complex$ and eigenvalue of $A$ an $n\times n$ matrix,
      define $X$ is an  eigenvector of $A$ associated to $\lambda$.
\item For $A$ an $n\times n$ matrix, define the characteristic polynomial
         of $A$.
\item For $A,B$ two $n\times n$ matrices, define $A$ is similar to $B$.
\item For $A$ an $n\times n$ matrix,  define $A$ is diagonalizable.
\end{enumerate}

\newpage

\prob Suppose $P_{n\times n}$ is an invertible matrix and
$L:\mat_{n\times n}\to \mat_{n\times n}$ is
defined by $L(A)=PAP^{-1}$ for any $A\in\mat_{n\times n}$.
Prove
that
\begin{enumerate}
  \item $L$ is a linear transformation,
  \item $L(AB)=L(A)L(B)$ for every $A,B\in \mat_{n\times n}$,
  \item $L$ is one-to-one,
  \item $L$ is onto,
  \item give the definition of $L^{-1}$. (For this particular $L$ not
  the general definition of inverse)
\end{enumerate}

\prob Suppose that $A$ is similar to $B$.  Prove that $A$ and
$B$ have the same characteristic polynomial.

\bigskip

\hrule

\bigskip

Extra Credit Optional Take Home or Do-it-now Problem:

\medskip

\noindent Suppose $A$ is an $n\times n$ matrix such that $A^2=Z$.
 Prove that $$\dim(\range(A))\leq {n\over 2}.$$

\bigskip

\noindent Submit your solution via email to miller@math.wisc.edu,
the earlier the better.


\end{document}