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\title{Engineering and Science Mathematics 2B}
\author{Homework 2}
\date{due February 14, 2003}
\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 A matrix $A$ is called normal if $AA^H=A^HA$, where $A^H$
denotes the Hermitian conjugate of $A$.  Show (a) that
$(A^H)^{-1}=(A^{-1})^H$, and (b) that $A^{-1}$ is normal if $A$ is
normal.

\item Let
\[
  B =
  \begin{pmatrix}
  0 & -i & i \\
  i & 0 & -i \\
  -i & i & 0 
  \end{pmatrix} \,,
  \qquad C= \frac1{\sqrt{8}}
  \begin{pmatrix}
    \sqrt3 & -\sqrt2 & -\sqrt3 \\
    1 & \sqrt6 & -1 \\
    2 & 0 & 2
  \end{pmatrix}\,.
\]
Are these matrices real, diagonal, symmetric, antisymmetric, singular,
orthogonal, Hermitian, anti-Hermitian, unitary, and/or normal?

\item 
\begin{itemize}
\item[(E)] Let
\[
  A =
  \begin{pmatrix}
  1 & 0 \\ 0 & 0
  \end{pmatrix}\,, \qquad
  B =
  \begin{pmatrix}
  0 & 0 \\ 3 & 4
  \end{pmatrix} \,.
\]
Compute $AB$ and $BA$.  Conclude that (i) $AB \neq BA$, and (ii) that
$AB=0$ does not imply that either $A$ or $B$ is the zero matrix.

\item[(A)] Prove that $AB=0$ implies that at least one of the matrices
is singular.
\end{itemize}

\item 
\begin{itemize}
\item[(E)] Determine if the following vectors are linearly dependent
or linearly independent:
\[
\begin{pmatrix}
  -4 \\ 0 \\ 1 \\ 5
\end{pmatrix}\,, \quad
\begin{pmatrix}
  -3 \\ -1 \\ 0 \\ 4
\end{pmatrix}\,, \quad
\begin{pmatrix}
  0 \\ 4 \\ 3 \\ 6
\end{pmatrix}\,.
\]

\item[(A)] Prove the following statement: Let $\v_1, \dots, \v_n$ be
linearly independent.  If a vector $\w$ can be written 
\[
  \w = \sum_{k=1}^n \alpha_k \, \v_k \,,
\]
the choice of the coefficients $\alpha_1,\dots,\alpha_n$ is unique.
\end{itemize}

\item Solve the following system of linear equations using the method
taught in class.
\begin{align*}
  x_1 + 3 \, x_2 - 5 \, x_3 &= 4 \\
  x_1 + 4 \, x_2 - 8 \, x_3 &= 7 \\
  -3 \, x_1 - 7 \, x_2 + 9 \, x_3 &= -6
\end{align*}

\item
\begin{itemize}
\item[(E)] Solve the following system of linear equations using the method
taught in class.
\begin{align*}
  x_1 + 3 \, x_2 +  x_3 &= 1 \\
  -4 \, x_1 -9  \, x_2 +2 \, x_3 &= -1 \\
  -3  \, x_2 -6 \, x_3 &= -3
\end{align*}
  
\item[(A)] Find conditions on $\alpha$ such that following system of
linear equations has (a) exactly one solution, (b) no solutions, or
(c) an infinite number of solutions; give all solutions where they
exist.
\begin{align*}
  x_1 + \alpha \, x_2  &= 1 \\
  x_1 - x_2 + 3 \, x_3 &= -1 \\
  2 \, x_1 -2 \, x_2 + \alpha \, x_3 &= -2
\end{align*}
\end{itemize}

\end{enumerate}

\end{document}
