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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Midterm II Retake}
\date{April 22, 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 $\displaystyle
A =
\begin{pmatrix}
  -\tfrac13 & \tfrac23 & 0 \\
  \tfrac23 & 0 & \tfrac23 \\
  0 & \tfrac23 & \tfrac13
\end{pmatrix} \,.$
\begin{enumerate}
\item Find the eigenvalues and eigenvectors of $A$.
\item For subspaces $U$ and $V$ of $\R^n$ we write $U = V^\bot$
if and only if $\operatorname{span} \{U,V\}=\R^n$ and $u^Tv=0$ for all
$u \in U$ and $v \in V$. 

Show that $\operatorname{Ker} A = (\operatorname{Range} A)^\bot$.
\item Do you expect the result of part (b) to be true for any matrix
$A \in M(3 \times 3)$?  Explain why or why not.
\end{enumerate}
\hfill(10+5+5)

%\newpage\mbox{}\newpage

\item Find an orthonormal basis for $P_2$, the vector space of real
polynomials of degree less or equal to two, endowed with inner product
\[
  \langle f, g \rangle = \int_0^1 f(x) \, g(x) \, \d x \,.
\]
\hfill(10)


%\newpage

\item Let $f$ be an odd function, i.e., $f(-x) = -f(x)$.
Show that $\tilde f(0)=0$.
\hfill(10)

%\newpage 

\item Let $f(x) = \sin^2 x \, \cos^2 x$.

\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
eigenvectors corresponding to distinct eigenvalues are
orthogonal. \mbox{}\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 the eigenvectors corresponding to distinct eigenvalues of a
skew-Hermitian map are orthogonal.  \hfill(10)
\end{itemize}

%\newpage

\item Let $f$ and $g$ be $2\pi$-periodic complex-valued functions
whose complex Fourier coefficients are denoted $f_k$ and $g_k$,
respectively.  Write
\[
  (f \otimes g) (x) = \int_0^{2\pi} f(y)^* \, g(x+y) \, \d y
\]
to denote their cross-correlation function.  Show that the Fourier
coefficients of $f \otimes g$ are given by $2 \pi \, f_k^* \, g_k$.
\hfill(10)
%\newpage

\item 
Compute the Fourier transform of $f(x) = 1+H(x) \, \e^{-x}$,
where $H$ denotes the Heavyside function.
\hfill(10)

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


\end{document}
