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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Midterm II}
\date{April 17, 2013}
\maketitle

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\noindent
For questions with an easy (E) and an advanced (A) version, choose
either the easy or the advanced version, not both.  You may make a
different choice for each question.


%\vspace*{\fill}
\noindent
Useful identities:
\begin{gather*}
  \cos \theta = \frac{e^{i\theta}+e^{-i\theta}}{2} \qquad
  \sin \theta = \frac{e^{i\theta}-e^{-i\theta}}{2i} \\
  \cos^2 \theta = \tfrac12 \, (1 + \cos 2\theta) \qquad
  \sin^2 \theta = \tfrac12 \, (1 - \cos 2\theta) \vphantom\int \\
  \cos \theta \, \sin \theta = \tfrac12 \, \sin 2 \theta \vphantom\int \\
  \tilde f(\xi)
  = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{-\i\xi x} \, f(x) \,
  dx \qquad 
  f(x)
  = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{\i\xi x} \,
  \tilde f(\xi) \, d\xi \\
  \delta(x)
  = \frac1{2\pi} \int_{-\infty}^\infty \e^{\i\xi x} \, d\xi  \\
  c_k = \frac1L \int_{-L/2}^{L/2} \e^{-2 \pi \i k x/L} \, f(x) \, \d x
  \qquad
  f(x) = \sum_{k=-\infty}^\infty c_k \, \e^{2 \pi \i k x/L}
\end{gather*}

%\newpage 

\begin{enumerate}

\item Let $\displaystyle
A =
\begin{pmatrix}
  1 & 2 & -1 \\
  2 & 4 & -2 \\
  -1 & -2 & 1
\end{pmatrix} \,.$
\begin{enumerate}
\item Find the eigenvalues and eigenvectors of $A$.
\item Find an orthonormal basis for $\R^3$ consisting of eigenvectors
of $A$.  
\item State a general result which guarantees that the computation in
(b) can be successfully completed.
\end{enumerate}
\hfill(10+5+5)

%\newpage\mbox{}\newpage

\item
\begin{itemize}
\item[(E)] Consider $P_2$, the vector space of polynomials of degree
less or equal to two, endowed with inner product
\[
  \langle f, g \rangle = \int_0^1 f(x) \, g(x) \, \d x \,.
\]
Show that the polynomials $p_1(x) = x-1$ and $p_2(x) = 4 \, x^2 + x
-1$ are orthogonal.  \mbox{}\hfill(8)
\item[(A)] Let $V$ be an $n$-dimensional complex vector space with inner
product $\langle \cdot, \cdot\rangle$ and orthonormal basis $\ee_1,
\dots, \ee_n$.  Let $\u,\v \in V$ have coordinate vectors $\a, \b \in
\C^n$ with respect to this basis.  Show that
\[
  \langle \u,\v \rangle = \a^H M \b
\]
where the matrix $M$ is Hermitian and its matrix elements are given by
\[
  m_{ij} = \langle \ee_i, \ee_j \rangle \,.
\] 
\hfill(10)
\end{itemize}

%\newpage

\item Show that the Fourier transform of an odd function is odd.

(Recall that a function $f$ is odd if $f(-x) = -f(x)$.)
\hfill(10)

%\newpage 

\item Let $f(x) = \sin x \, \cos x$.

\begin{itemize}
\item[(E)] Compute the Fourier series of $f$ on the interval $[0,2\pi]$.
  \hfill(8)
\item[(A)] Compute the Fourier transform of $f$ on  $\R$.
  \hfill(10)
\end{itemize}

%\newpage

\item 
\begin{itemize}
\item[(E)] Show that if a matrix $S$ is real symmetric, then its
eigenvalues are real. \hfill(8)
\item[(A)] Let $V$ be a complex vector space with inner product
$\langle \cdot, \cdot \rangle$.  We say that a linear map $L \colon V
\to V$ is \emph{skew-Hermitian} provided
\[
  \langle Lv, w \rangle = - \langle v, Lw \rangle 
\]
for all $v,w \in V$.  

Show that if $\lambda$ is an eigenvalue of a skew-Hermitian map, then
$\Re \lambda = 0$.  \hfill(10)
\end{itemize}

%\newpage

\item 
\begin{itemize}
\item[(E)] Let $c_k$ denote the complex Fourier coefficients of a
$2\pi$-periodic function $f$.  Show that the Fourier coefficients of $f'$ are
given by $\i k \, c_k$.  \hfill(8)
\item[(A)] Consider the derivative operator $Lf = f'$ as a linear map
on the vector space of smooth periodic functions on the interval
$[0,2\pi]$ endowed with inner product
\[
  \langle f, g \rangle = \frac1{2\pi} \int_0^{2\pi} f^*(x) \, g(x) \,
  \d \x \,.
\]
Show that $L$ is skew-Hermitian in the sense of Question~5A.  Relate
this fact to what you already know about the eigenvalues of the
derivative operator.
\hfill(10)
\end{itemize}

%\newpage

\item Compute the Fourier transform of $f(x) = \e^{-\lvert x
\rvert}$. 
\hfill(10)

%\newpage\mbox{}\newpage
\end{enumerate}


\end{document}
