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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 5}
\date{due March 15, 2003, before 22:00}
\maketitle

\noindent
Normal questions and advanced questions (A) are worth $5$ points; easy
questions (E) are worth $4$ points.  Complete either the easy, or the
advanced version, not both.

\begin{enumerate}
\item Using the Gram--Schmidt procedure, construct an orthonormal set
of vectors from
\[
  \v_1 = 
  \begin{pmatrix}
  0 \\ 0 \\ 1 \\ 1 
  \end{pmatrix} \,,
  \quad
  \v_2 = 
  \begin{pmatrix}
  1 \\ 0 \\ -1 \\ 0
  \end{pmatrix} \,,
  \quad
  \v_3 = 
  \begin{pmatrix}
  1 \\ 2 \\ 0 \\ 2
  \end{pmatrix} \,,
  \quad
  \v_4 = 
  \begin{pmatrix}
  2 \\ 1 \\ 1 \\ 1
  \end{pmatrix} \,.
\]

\item 
\begin{itemize}
\item[(E)] Using the Gram--Schmidt procedure, find an orthonormal
basis for the subspace of $\R^3$ spanned by
\[
  \v_1 = 
  \begin{pmatrix}
  1 \\ 2 \\ 3
  \end{pmatrix} \,,
  \quad
  \v_2 = 
  \begin{pmatrix}
  4 \\ -1 \\ 0
  \end{pmatrix} \,.
\]


\item[(A)] Let $\{ \w_1,\dots,\w_m\}$ be an orthonormal set of vectors
in a vector space $V$ with an inner product.  Bessel's inequality (see
p.\ 351 in the book) states that for every vector $\x \in V$,
\[
  \lVert \x \rVert^2 \geq
  \sum_{j=1}^m \lvert \langle \x, \w_j \rangle \rvert^2 \,.
\]
Prove that Bessel's inequality becomes an equality if and only if
\[
  \x\in \Span \{ \w_1, \dots, \w_m \} \,.
\]

\end{itemize}

\item A matrix $A$ is called unitary if $A^H=A^{-1}$.  Explain why
this is the same as saying that the columns of $A$ are orthonormal, or
that the rows of $A$ are orthonormal.

\item
\begin{itemize}
\item[(E)] Find the eigenvalues and eigenvectors of 
\[
  A = \begin{pmatrix}
  1 & -1 & 0 \\
  -1 & 1 & 0 \\
  0 & 0 & 1
\end{pmatrix} 
\]
and show that the eigenvectors are orthogonal.

\item[(A)] Let 
\[
  A = \begin{pmatrix}
  1 & \alpha & 0 \\
  \beta & 1 & 0 \\
  0 & 0 & 1
\end{pmatrix} \,,
\]
where $\alpha$ and $\beta$ are non-zero complex numbers.  Find its
eigenvalues and eigenvectors.  Find conditions for (a) the eigenvalues
to be real and (b) the eigenvectors to be orthogonal.  Show that the
conditions are jointly satisfied if and only if $A$ is Hermitian.

\end{itemize}


\item Let $P_2$ be the vector space of polynomials with real
coefficients of degree less or equal to $2$, and consider the inner
product
\[
  \langle p,q \rangle = \int_0^1 p(x) \, q(x) \, dx \,.
\]
\begin{itemize}
\item[(E)] Apply the Gram--Schmidt procedure to the basis
$\{1,x,x^2\}$. 

\item[(A)] In addition to (E), explain \emph{briefly} why $\langle \,
\cdot \,, \, \cdot \, \rangle$ is an inner product.

\end{itemize}

\item Consider the vector space of continuous complex-valued functions
on $[0,2 \pi]$ with inner product
\[
  \langle f,g \rangle = \int_0^{2\pi} f^*(x) \, g(x) \, dx \,.
\]
Show that the functions $f(x)=\e^{ikx}$ and $g(x)=\e^{ijx}$ are
orthogonal when $k \neq j$.

\end{enumerate}

\end{document}
