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\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 10/31/08}\\
{\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:} To solve a system of equations most
  efficiently, the matrix should be reduced to echelon, but {\em not}
  to reduced echelon (canonical) form.

  {\bf Question:} Reduce the matrix below to echelon form but not
  reduced echelon form using class procedures.  Do not do more
  operations than needed.
  \begin{displaymath}
    \left(
      \begin{array}{cccc}
        0 & 0 & 1 \\
        0 & 1 & 2 \\
        1 & 2 & 3 \\
        4 & 5 & 6 \\
        7 & 8 & 9
      \end{array}
    \right)
  \end{displaymath}

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} For purposes such as analyzing natural
  frequencies, sometimes an analytical expression of a determinant is
  needed.

  {\bf Question:} Find, without {\em any} row (or column) operations
  \begin{displaymath}
    \left|
      \begin{array}{cccccc}
        0 & 0 & 1 & 0 & 0 & 0 \\
        0 & 0 & 2 & 3 & 0 & 4 \\
        0 & 0 & 5 & 0 & 0 & 6 \\
        7 & 0 & 8 & 9 & 0 & 1 \\
        2 & 0 & 3 & 4 & 6 & 5 \\
        7 & 8 & 9 & 1 & 3 & 2 
      \end{array}
    \right|
  \end{displaymath}
  At every stage, choose the approach that requires the smallest
  possible number of terms.  If you do it correctly, it will be quick.

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Streamlines in the vicinity of stagnation
  points can often be found as lines on which a quadratic form is
  constant.

  {\bf Question:} Using class procedures, find and very accurately and
  neatly draw the lines on which $6x_1^2-12x_1x_2+x_2^2=-3$.  List all
  relevant angles and intercepts in the picture to fully define it.

\begin{rawhtml}
<A HREF="figures/3.jpg">Solution.</A>
\end{rawhtml}

\end{enumerate}

\end{document}
