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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due March 2
    (Friday) }} \rhead{\thepage} \cfoot{{\sf MthSc 434 $|$
    Advanced Engineering Mathematics $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}

\item For this problem, consider the vector space $V=\R^3$ and use the
  vector dot product as the inner product.
  \begin{enumerate}
  \item Show that the set of vectors $\{v_1,v_2,v_3\}$, where 
    \[
    v_1=(1,2,-2)\,,\qquad v_2=(0,1,1)\,,\qquad v_1=(-4,1,-1)\,.
    \]
    is an orthogonal set, but not orthonormal. 
  \item Normalize $v_1$, $v_2$, and $v_3$ to get an
    \emph{orthonormal} basis of $\R^3$. That is, compute the
    following:
    \[
    \mathcal{B}=\{n_1,n_2,n_3\}\,\qquad\mbox{where}\quad
    n_i=\frac{v_i}{||v_i||}\,.
    \]
  \item Use the dot product to express the vector $w=(1,2,3)$ in terms
    of $n_1$, $n_2$, and $n_3$. That is, find $C_1$, $C_2$, and $C_3$
    such that
    \[
    w=C_1n_1+C_2n_2+C_3n_3\,.
    \]
  \end{enumerate}


\item Let $\Poly_3(\R)=\{a_3x^3+a_2x^2+a_1x+a_0\mid a_i\in\R\}$, the
  vector space of polynomials of degree at most $3$. Define the
  following \emph{inner product} on $\Poly_3(\R)$:
  \[
  \<f,g\>=\int_{-1}^1 f(x)g(x)\,dx\,.
  \]
  \begin{enumerate}
    \item Verify that this is indeed an inner product on
      $\Poly_3(\R)$.
    \item Consider the two sets
      \[
      \mathcal{B}_1=\{1,x,3x^2-1,5x^3-3x\}\,,\qquad
      \mathcal{B}_2=\{1,x,x^2,x^3\}
      \]
      that are both bases for $\Poly_3(\R)$. Show that $\mathcal{B}_1$
      is an orthogonal set, but $\mathcal{B}_2$ is not. (The set
      $\mathcal{B}_1$ are the first four \emph{Legendre polynomials},
      $P_n(x)$ for $n=0,\dots,3$. When we study Sturm-Liouville
      theory, we will see why the Legendre polynomials are always orthogonal!)
    \item For each $f\in\mathcal{B}_1$, compute the \emph{norm} of
      $f$, which is defined as $||f||=\<f,f\>^{1/2}$. Find an
      \emph{orthonormal} basis for $\Poly_3(\R)$ by normalizing the
      elements in $\mathcal{B}_1$.
    \item Consider the polynomial $f(x)=3x^3-2x^2+4$. Use
      orthogonality to write $f(x)$ using the elements in
      $\mathcal{B}_1$. That is, find $C_0$, $C_1$, $C_2$, and $C_3$
      such that
      \[
      3x^3-2x^2+4=C_0+C_1x+C_2(3x^2-1)+C_3(5x^3-3x).
      \]
  \end{enumerate}


\item Find the Fourier series of the following functions
  \emph{without} computing any integrals.
  \begin{enumerate}
  \item $f(x)=2-3\sin 4x+5\cos 6x$,
  \item $f(x)=\sin^2 x$. [\emph{Hint}: Use a standard trig identity.]
  \end{enumerate}


\item Consider the $2\pi$-periodic function defined by
  \[
  f(x)=\left\{\begin{array}{ccc} x^2 && -\pi\leq x<\pi, \\
  f(x-2k\pi) && -\pi+2k\pi\leq x<\pi+2k\pi. \end{array}\right.
  \]
  Sketch this function on $[-7\pi,7\pi]$ and compute its Fourier
  series. Feel free to use a computer to evaluate the indefinite
  integral $\int x^2\cos nx\,dx$.


\item Let $\Per_{2\pi}(\C)$ be the set of piecewise continuous
  $2\pi$-periodic complex-valued functions, and define the (complex)
  inner product
  \[
  \<f,g\>=\frac{1}{2\pi}\int_{-pi}^\pi f(x)\overline{g(x)}\,dx\,,
  \]
  where $\overline{g(x)}$ denotes the complex conjugate of $g(x)$.
  \begin{enumerate}
    \item Verify that this is indeed a complex inner product on
      $\Per_{2\pi}(\C)$.
    \item Verify that $\mathcal{B}=\{e^{inx}\mid n\in\Z\}$ is an
      orthonormal set with respect to this inner product. That is,
      show that for each $f,g\in\mathcal{B}$, we have $\<f,g\>=1$ if
      $f=g$ and $0$ if $f\neq g$. [\emph{Hint}: Recall that
        $\overline{e^{inx}}=e^{-inx}$.]
    \item Given the additional knowledge that $\mathcal{B}$ is an
      orthonormal \emph{basis} of $\Per_{2\pi}(\C)$ (you may assume
      this), we can write any $f\in\Per_{2\pi}(\C)$ uniquely as
      \[
      f(x)=\sum_{n=-\infty}^\infty c_ne^{inx}\,,\qquad c_n\in\C\,.
      \]
      Use the results from Parts (a) and (b) to derive a formula for
      each $c_n$. [\emph{Hint}: Think of $c_n$ as the magnitude of the
        ``projection'' of $f(x)$ onto $e^{inx}$.]
  \end{enumerate}


  
\end{enumerate}

\end{document}
