\documentclass{article}

% paper size. [11in]
%\setlength{\paperheight}{11in}
% top margin (1 inch is added automatically) [17pt]
\setlength{\topmargin}{-24pt}
% height of the header line [12pt]
%\setlength{\headheight}{12pt}
% separation between header and body [25pt]
\setlength{\headsep}{12pt}
% from start of body to the first base line [12pt]
%\setlength{\topskip}{12pt}
% size of the body of the page (no headers or footers) [7.618in] 
\setlength{\textheight}{9truein}
% bottom of body to base of footer [30pt]
\setlength{\footskip}{24pt}

% paper size. [8.5in]
%\setlength{\paperwidth}{8.5in}
% left margin (1 inch is added automatically) [39pt]
\setlength{\oddsidemargin}{0truein}
% the same for left-hand pages [40pt]
\setlength{\evensidemargin}{0truein}
% size of the body of the page [5.417in] 
\setlength{\textwidth}{6.5truein}
% width and separation of marginal notes
\setlength{\marginparwidth}{.8truein}\setlength{\marginparsep}{.1truein}

\pagestyle{empty}

%\setlength{\parindent}{0pt}\setlength{\parskip}{\baselineskip}

%\renewcommand{\baselinestretch}{2}

%\setlength{\jot}{9pt}

\usepackage{html}
\usepackage{epsf}
%\usepackage{cite}

%\bibliographystyle{plain}

\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 10/30/09}\\
{\it Closed book \hfill Van Dommelen \hfill 11:50-12:40 pm}
\end{center}

Solutions should be fully {\em derived} showing all intermediate
results, using class procedures.  Show all reasoning.  Bare answers
are absolutely not acceptable, because I will assume they come from
your calculator (or the math handbook, sometimes,) instead of from
you.  You must state what result answers what part of the question if
there is any ambiguity.  Answer exactly what is asked; you do not get
any credit for making up your own questions and answering those.  Use
the stated procedures.  Give exact, cleaned-up, answers where
possible.

You {\em must} use the systematic procedures described in class, not
mess around randomly until you get some answer.  Geometry must be done
using vector operations.  Use standard matrix methods to determine
linear independence and simplified bases of vector spaces.  You need
to reduce matrices to echelon form where elimination is called for,
using the basic row operations and following the class procedures
exactly.  Do not take shortcuts.  Do not reduce further if there is no
need.  Eigenvalues must be found using minors only.  Eigenvectors must
be found by identifying the basis vectors of the appropriate null
space if there are multiple eigenvalues, using the appropriate
procedures.  Eigenvectors to symmetric matrices must be orthonormal.
Higher matrix powers and polynomials must be found through
transformation, not crunching.  Inverses must be found the quick way,
where possible.

One book of mathematical tables, such as Schaum{\rq}s Mathematical
Handbook, may be used, as well as a calculator, and a handwritten
letter-size formula sheet.

\begin{enumerate}
  
\item {\bf Background:} Finding the possible internal stress fields in
  structures requires finding the null space of the matrix of the
  governing equilibrium equations.

  {\bf Question:} (33\%) For the matrix
  \begin{displaymath}
    M = \left(
      \begin{array}{ccccc}
        0 & 0 & 4 & 6 & 8 \\
        0 & 2 & 4 & 6 & 8 \\
        0 & 1 & 3 & 5 & 7 \\
        0 & 1 & 0 & 0 & 0
      \end{array}
    \right)
  \end{displaymath}
  (a) find the null space, (b) find a basis for the null space, (c)
  find the dimension of the null space, (d) find the rank.  Identify
  what is what.

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} (33\%) The matrix below will also be used in
  question 3, so don{\rq}t make mistakes:

  {\bf Question:} Find an orthonormal transformation matrix that
  diagonalizes the matrix
  \begin{displaymath}
    A =
    \left(
      \begin{array}{cc}
        2 & 3 \\
        3 &10
      \end{array}
    \right)
  \end{displaymath}
  Find the inverse of the transformation matrix.  Be sure to use the
  class procedures for a matrix of the type of $A$.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Transformations as the one above are very
  useful in simplifying complex tasks, like finding powers of matrices
  and analyzing the transition to instability of dynamical systems.

  {\bf Question:} For the matrix $A$ given in question 2:
  \begin{enumerate}
  \item (17\%) Find a cube root of the matrix.
\begin{rawhtml}
<A HREF="figures/3a.jpg">Solution.</A>
\end{rawhtml}
  \item (17\%) Accurately draw the quadratic curve
    $\vec{x}^TA\vec{x}=1$.  Show angles and intercepts to scale the
    curve properly.

    Then in separate graphs sketch the corresponding curves for
    \begin{displaymath}
      A_2 =
      \left(
        \begin{array}{cc}
          1 & 3 \\
          3 & 9
        \end{array}
      \right)
      \qquad
      A_3 =
      \left(
        \begin{array}{cc}
          0 & 3 \\
          3 & 8
        \end{array}
      \right)
    \end{displaymath}
    This requires very little additional work, since they have the
    same eigenvectors as $A$.  However, $A_2$ has eigenvalues that are
    1 less than those of $A$, while the eigenvalues of $A_3$ are 2
    less than those of $A$
\begin{rawhtml}
<A HREF="figures/3b.jpg">Solution.</A>
\end{rawhtml}
  \end{enumerate}

\end{enumerate}

\end{document}
