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\noindent
Math 261

\begin{center}
{\bf Study Guide for Final}
\end{center}

\bigskip


\noindent
Due at 8:00 a.m., December 16, 2001

\bigskip

\noindent
Create a study guide for the course.  It should include key
definitions
and theorems and a brief outline of important techniques.  It should
also indicate relations among different parts of the course (for
example,
how solutions of systems of equations are
related to subspaces and dimensions).  You may include other things
you consider important for understanding the course.  Your study guide
should be organized in a reasonable way.  This will be treated as 
a 50 point homework assignment.  In particular, you are free to
discuss it
 with
anyone, but you must write it yourself.   


\begin{center}
{\bf Take-home problems for the Final}
\end{center}

\bigskip

\noindent
Due at 8:00 a.m., December 16, 2001

\bigskip

\noindent
You may consult your course notes, homework, Hoffman and Kunze and 
ATLAST.
You may use Maple, Matlab or Mathematica as an experimental
tool  on any part of the problem.  If you do, please indicate how on your exam.
You may not consult any other books or notes.  You may not discuss
the exam with anyone except me.

\bigskip

\noindent
Be sure to explain your answers.  For example, if you are asked
to find a basis for a certain vector space, I expect you not only
to find it but also to prove that it is a basis.  Make sure that your
explanations are clear.  The word \emph{show}
is a synonym for  \emph{prove}.
 
\bigskip

\begin{enumerate}




\item
Let 
\begin{math}
  A
\end{math}
be an
\begin{math}
  m\times n
\end{math}
matrix over a field
\begin{math}
  F.
\end{math}
Suppose
\begin{math}
  \mbox{rank} (A)=r
\end{math}
and the
system of equations 
\begin{math}
  AX=Y
\end{math}
has a  solution
\begin{math}
  X_0.
\end{math}
Describe all solutions to the system.
(\emph{Hint}: Suppose  
\begin{math}
  X_1
\end{math}
is another solution.
What can you say about 
\begin{math}
  X_0-X_1?)
\end{math}





\bigskip

\item
\begin{enumerate}
\item Let 
  \begin{math}
  A,B\in F^{n\times n}. 
  \end{math}
Prove that 
\begin{math}
  \tr(AB)=\tr (BA).
\end{math}
(The trace 
\begin{math}
  \tr
\end{math}
of a matrix is defined in Example 19 on page 98 of Hoffman
and Kunze.)

\medskip

\item Suppose 
  \begin{math}
    A
  \end{math}
and 
\begin{math}
  B
\end{math}
are similar.  Show that 
\begin{math}
  \tr(A)=\tr(B).
\end{math}

\medskip

\item Let 
  \begin{math}
    V
  \end{math}
be a finite dimensional vector space, 
\begin{math}
  T:V\to V
\end{math}
a linear operator.  Let 
\begin{math}
 \mathcal B, \mathcal B'
\end{math}
be  ordered bases of V.
Show that 
\begin{math}
  \tr [T]_\mathcal B = \tr [T]_{\mathcal B'}.
\end{math}
 Use this to define the \textbf{trace} of
 \begin{math}
   T,\ \tr T.
 \end{math}





\item Let 
  \begin{math}
    V
  \end{math}
be the space of all 
\begin{math}
  2\times 2
\end{math}
matrices over a field 
\begin{math}
  F
\end{math}
and let 
  \begin{math}
    P\in V
  \end{math}
be fixed.  Define a linear operator 
\begin{math}
  T
\end{math}
on 
\begin{math}
  V
\end{math}
by 
\begin{math}
  T(A)=PA.
\end{math}
Show that 
\begin{math}
  \tr T=2\,\tr P.
\end{math}

\end{enumerate}

\bigskip
\item
Let 
\begin{math}
  V
\end{math}
be the space of all real polynomials and let 
\begin{math}
  f
\end{math}
be the linear functional on 
\begin{math}
  V
\end{math}
defined by 
\begin{math}
  f(p)=\int _0 ^1 p(x)\,dx.
\end{math}
Let 
\begin{math}
  D:V\to V
\end{math}
be differentiation.  What is 
\begin{math}
  D^t f?
\end{math}

\bigskip
\item 
\begin{enumerate}
\item
 Prove:
If 
\begin{math}
  A
\end{math}
is an 
\begin{math}
  n\times n
\end{math}
matrix over 
\begin{math}
  \mathbf C
\end{math}
there are at most 
\begin{math}
  n
\end{math}
distinct scalars 
\begin{math}
  c
\end{math}
such that 
\begin{math}
  \det (cI-A)=0.
\end{math}

\medskip

\noindent
\item
Give an example where there are 
\begin{math}
  n
\end{math}
distinct scalars as in (a).


\medskip
\noindent
\item Show that if 
\begin{math}
  \det (cI-A)=0
\end{math}
there is an 
\begin{math}
  x\in \mathbf C ^n,\ x\ne 0
\end{math}
such that 
\begin{math}
  Ax=cx.
\end{math}
\end{enumerate}

\bigskip

\item
 An 
\begin{math}
  n\times n
\end{math}
matrix over 
\begin{math}
  \mathbf R
\end{math}
is \emph{orthogonal}
if 
\begin{math}
  A^t A = I.
\end{math}

\medskip
\begin{enumerate}

\item Show that if 
\begin{math}
  A
\end{math}
is an 
\begin{math}
  n\times n
\end{math}
orthogonal
matrix over 
\begin{math}
  \mathbf R
\end{math}
then 
\begin{math}
  \det A = \pm 1.
\end{math}

\medskip
\noindent

\item Give an example of a 
\begin{math}
  2\times 2
\end{math}
orthogonal matrix 
\begin{math}
  A
\end{math}
over 
\begin{math}
  \mathbf R
\end{math}
such that 
\begin{math}
  \det A = -1.
\end{math}
\end{enumerate}

\bigskip

\noindent
\item

\begin{enumerate}

\item Show that if the entries of
\begin{math}
  A
\end{math}
are integers and 
\begin{math}
  \det A = \pm 1
\end{math}
then the entries of 
\begin{math}
  A^{-1}
\end{math}
are integers.

\medskip

\noindent
\item
 Suppose the entries of  
\begin{math}
  A
\end{math}
are integers and 
\begin{math}
  A
\end{math}
is nonsingular.
 What can you say about the entries of 
 \begin{math}
   A^{-1}
 \end{math}?


\medskip
\noindent
\item Show that if the entries of 
\begin{math}
  A
\end{math}
and
\begin{math}
   A^{-1}
\end{math}
are integers then 
\begin{math}
  \det A = \pm 1.
\end{math}

\end{enumerate}
\end{enumerate}
\end{document}

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