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\begin{center}
{\LARGE {Mathematics 118A, Winter 1998}}
\end{center}
\begin{center}
\large Midterm Exam I
\end{center}
\bigskip

\begin{enumerate}

\item The norm of a linear transformation $A \colon \R^n \to \R^n$ is
defined
\begin{equation*}
  \| A \| = \sup_{\substack{x \in \R^n \\ x \neq 0}}
            \frac{\| Ax \|}{\| x \|}.
\end{equation*}
\begin{enumerate}
\item Prove that $\| A+B \| \leq \| A \| + \| B \|$.
\item Show that one can also write
$ \displaystyle
  \| A \| = \sup_{ \|x\|=1}
            \| Ax \|.
$
\item Compute $\| A \|$ for
$ \displaystyle
  A = \begin{pmatrix}
        1 & 0 \\ 0 & 2 
      \end{pmatrix}.
$

Hint: Use part (b).  How can you write any unit vector in $\R^2$?
\hfill (4+3+3)
\end{enumerate}



\item Compute the matrix exponential $\exp(At)$ for the matrix
\begin{equation*}
  A = \begin{pmatrix}
        -1 & -1 \\ 1 & -3 
      \end{pmatrix}.
\end{equation*}
Hint: The Jordan normal form of $A$ is $A = TJT^{-1}$ with
\begin{equation*}
  T = \begin{pmatrix}
        1 & 0 \\ 1 & -1 
      \end{pmatrix}
  \quad \mbox{and} \quad
  J = \begin{pmatrix}
        -2 & 1 \\ 0 & -2 
      \end{pmatrix}.
\end{equation*}
What is the relationship between $\exp(At)$ and $\exp(Jt)$?
\hfill(10)

\item Find all equilibrium points of the system
\begin{align*}
  \dot x &= y - x^2 \\
  \dot y &= 1 - y 
\end{align*}
and determine their stability.  Sketch the phase portrait.  (You may
wish to compute some of the eigenvectors of the linearization to
determine the directions of stable and unstable manifolds.)
\hfill(10)

\item Consider the second order equation
\begin{equation*}
  \ddot x = - x^3.
\end{equation*}
\begin{enumerate}
\item Find the equilibrium point of the flow. (There is only one.)
\item Show that the linear analysis of the stability of the critical
point is inconclusive.
\item Find a first integral corresponding to the energy of the system.
\item At the equilibrium point, does the energy have a local minimum,
a local maximum, or a saddle point?  What does this tell you about the
(nonlinear) stability of the system?
\item Is the flow volume preserving?
\hfill(2+2+2+2+2)
\end{enumerate}



\end{enumerate}
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