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\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 10/29/10}\\
{\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.  {\em 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.  {\em Avoid fractions where possible}. 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:} Null spaces can describe such things as internal
  forces in trusses.

  {\bf Question:} (34\%) Using class procedures find the null space of
  the matrix:
  \begin{displaymath}
    \left(
      \begin{array}{ccccc}
        1 & 2 & 3 & 4 & 5 \\
        6 & 7 & 8 & 9 & 0 \\
        3 & 5 & 7 & 9 & 0
      \end{array}
    \right)
  \end{displaymath}
  Find a basis to the null space and simplify it using class
  procedures.

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} To find inverses for small matrices,
  minors are often useful.

  {\bf Question:} (33\%) Using minors only, find the inverse of the
  orthogonal, but not orthonormal, matrix
  \begin{displaymath}
    \left(
      \begin{array}{rrr}
        1 & 1 & 1 \\
        1 & 0 &-2 \\
        1 &-1 & 1
      \end{array}
    \right)
  \end{displaymath}

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Quadratic forms commonly arise from systems that
  are linearized for small displacements from equilibrium.

  {\bf Question:} (33\%) Using class procedures, analyze and very
  accurately draw:
  \begin{displaymath}
    2 x^2 + 4 xy + 5 y^2 = 1
  \end{displaymath}
  Identify all angles in the graph.  Show the location of 1 on your
  axes.

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

\end{enumerate}

\end{document}
