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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Midterm II}
\date{March 31, 2004}
\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} \\
  \sin \theta = \frac{e^{i\theta}-e^{-i\theta}}{2i} \\
  \cos^2 \theta = \tfrac12 \, (1 + \cos 2\theta) \\
  \sin^2 \theta = \tfrac12 \, (1 - \cos 2\theta) \\
  \tilde f(\xi)
  = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty e^{-i\xi x} \, f(x) \,
  dx \\
  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 
\end{gather*}




\begin{enumerate}
\item 
\begin{itemize}
\item[(E)] Define an inner product on $\R^2$ by
\[
  \langle \u, \v \rangle = \u^T A \v
  \qquad \text{with} \qquad
  A =
  \begin{pmatrix}
  2 & 1 \\ 1 & 1
  \end{pmatrix} \,.
\]
Are the vectors
\[
  \v =
  \begin{pmatrix}
  1 \\ 1
  \end{pmatrix}
  \qquad \text{and} \qquad
  \w =
  \begin{pmatrix}
  1 \\ -1
  \end{pmatrix}
\]
orthogonal with respect to this inner product?  Explain.
\hfill (8)

\item[(A)] Show that $\langle \u, \v \rangle = \u^H A \v$ defines an
  inner product on $\C^n$ provided that $A$ is a Hermitian $n \times
  n$ matrix, and provided that all eigenvalues of $A$ are strictly
  positive.
\hfill (10)

\end{itemize}


\item
\begin{itemize}
\item[(E)] Find an orthonormal basis for the subspace of $\R^3$
spanned by
\[
  \v_1 =
  \begin{pmatrix}
  1 \\ 1 \\ 1
  \end{pmatrix} \,, \qquad
  \v_2 =
  \begin{pmatrix}
  1 \\ -1 \\ 1
  \end{pmatrix} \,.
\]
\hfill (8)

\item[(A)] Find an orthonormal basis for the subspace of $C[0,\pi]$
spanned by
\[
  f_1(x) = \sin^2 x \,, \qquad
  f_2(x) = \cos^2 x \,,
\]
with respect to the inner product
\[
  \langle f, g \rangle
  = \int_0^{2\pi} f(x) \, g(x) \, \d x \,.
\]
\emph{You do not need to explicitly compute the normalization of the
  second basis function.}
\hfill (10)
\end{itemize}


\item
\begin{itemize}
\item[(E)] Show that the Fourier coefficient with wavenumber $k=1$ of
  the function
\[
  f(x) =
  \begin{cases}
    x & \text{for } 0 \leq x < \pi \\
    x - \pi & \text{for } \pi \leq x < 2\pi
  \end{cases}
\]
is zero.
\hfill (8)

\item[(A)] Let $f$ be a $\pi$-periodic function on the interval $[0,
2\pi]$, i.e.\ $f(x) = f(x+\pi)$.  Show that the Fourier coefficient
$f_k=0$ if $k$ is odd.
  
\hfill (10)
\end{itemize}


\item
\begin{enumerate}
\item Derive the Parseval identity
\[
  \int_{-\infty}^{\infty} |f(x)|^2 \, \d x
  = \int_{-\infty}^{\infty} |\tilde f(\xi)|^2 \, \d\xi \,.
\]

\item Show that the Fourier transform of
\[
  f(x) =
  \begin{cases}
    1 & \text{for } -1 \leq x \leq 1 \\
    0 & \text{otherwise} \,.
  \end{cases}
\]
is given by
\[
  \tilde f (\xi)
  = \sqrt{\frac2\pi} \, \frac{\sin\xi}\xi \,.
\]

\item Use (a) and (b) to prove
\[
  \int_{-\infty}^{\infty} \frac{\sin^2 \xi}{\xi^2} \, \d\xi = \pi \,.
\]

\hfill (10+10+10)

\end{enumerate}



\item Let $f$ be a real-valued odd function, i.e. $f(-x)=-f(x)$.
\begin{enumerate}
\item Show that the Fourier transform $\tilde f$ is also odd.
\item Show that the Fourier transform $\tilde f$ is purely imaginary.
\end{enumerate}
\hfill (10+10)
\end{enumerate}


\end{document}
