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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Midterm I}
\date{March 6, 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.

%\newpage

\begin{enumerate}
\item
\begin{itemize}
\item[(E)] Let $A \in M(n \times n)$ be a regular matrix and suppose
that $\lambda$ is an eigenvalue of $A$.  Show that $1/\lambda$ is an
eigenvalue of $A^{-1}$.  \hfill(8)

\item[(A)] Let $A \in M(n \times n)$ be diagonalizable with eigenvalues
$\lambda_1, \dots, \lambda_n$ (not necessarily distinct) and $\v_1,
\dots, \v_n$ corresponding linearly independent eigenvectors.  How can
you choose, given this information, a basis for $\Range A$?  Explain
why your method works.
\hfill(10)
\end{itemize}
%\newpage


\item Let $\displaystyle A = 
\begin{pmatrix}
  0 & 3 \\ -3 & 0 
  % Version B: 0 & 2 \\ -2 & 0 
  % Version A: 0 & 1 \\ -1 & 0
\end{pmatrix} \,.$
\begin{enumerate}
\item Find the eigenvalues and eigenvectors of $A$.
\item Write out a  diagonal matrix $D$ and an invertible matrix
$S$ such that $D=S^{-1}AS$.
\item Check your result by explicitly performing the matrix
multiplications $SD$ and $AS$.  
\end{enumerate}
\hfill(10+5+5)
%\newpage

%\mbox{}\newpage

\item Let $\v=(1,1,-1)^T$.
% Version B: $\v=(1,-1,1)^T$.
% Version A: $\v=(-1,1,1)^T$.
\begin{itemize} 
\item[(E)] Find the distance of the point $\boldsymbol{p}=(0,1,0)^T$
to the line through the origin in the direction of $\v$.

%\hfill (8)

\item[(A)] Find the matrix representation in the standard basis of the
projection onto the plane through the origin which is normal to $\v$.
\hfill (10)

\end{itemize}
%\newpage

\item Find the general solution to the system of linear equations $A\x
= \b$ with
\begin{itemize}
\item[(E)]
\[
  A =
  \begin{pmatrix}
  1 & 7 & 4 & 4 \\
  0 & 6 & 3 & 3 \\
  1 & 1 & 1 & 1 \\
  \end{pmatrix} \,,  
  % Version B: A =
  % \begin{pmatrix}
  % 1 & 4 & 4 & 7 \\
  % 0 & 3 & 3 & 6 \\
  % 1 & 1 & 1 & 1 \\
  % \end{pmatrix} \,,  
  % Version A: \begin{pmatrix}
  % 4 & 4 & 1 & 7 \\
  % 3 & 3 & 0 & 6 \\
  % 1 & 1 & 1 & 1 \\
  % \end{pmatrix} \,,
  \qquad
  \b =
  \begin{pmatrix}
  -5 \\ -6 \\ 1
  \end{pmatrix} \,.
\] 
\hfill(16)

\item[(A)]
\[
  A =
  \begin{pmatrix}
  0  & 1+3i  & 3-i \\
  -1 & -1+5i & 2-i \\
  i  & 2+2i  & i   \\
  \end{pmatrix} \,,  
  % Version B: \begin{pmatrix}
  % 0  & 3-i & 1+3i \\
  % -1 & 2-i & -1+5i \\
  % i  & i   & 2+2i \\
  % \end{pmatrix} \,,  
  % Version A: \begin{pmatrix}
  % 3-i & 0  & 1+3i \\
  % 2-i & -1 & -1+5i \\
  % i   & i  & 2+2i \\
  % \end{pmatrix} \,,
  \qquad
  \b =
  \begin{pmatrix}
  10 \\ 7 \\ 3i
  \end{pmatrix} \,.
\]
\hfill (20)
\end{itemize}
Check your answer! (Required for full credit.)
%\newpage\mbox{}\newpage

\item Let $V$ be the set of symmetric $2 \times 2$ real matrices.
\begin{enumerate}
\item Show that $V$ is a vector space with the usual matrix addition
and scalar multiplication.  
\item Find a basis $B$ for the vector space $V$.  (Keep it simple, do
not use a basis containing the matrices $I, E, S$ from below!)
\item Show that $B' = \{I,E,S\}$ where $I$ is the identity matrix,
\[ 
  E=
  \begin{pmatrix}
    1 & 1 \\ 1 & 1
  \end{pmatrix} , \qquad \text{and} \qquad
  S =
  \begin{pmatrix}
    0 & 1 \\ 1 & -1
  \end{pmatrix}
\]
is another basis $V$. Compute the change of coordinate matrices
$I_{B',B}$ and $I_{B,B'}$.

\item Show that for any fixed $C \in M(2\times 2)$ the map $F$ defined
by $F(A) = CAC^T$ is a linear map on $V$.

\item Let
\[
  C =
  \begin{pmatrix}
    0 & -1 \\ 1 & 0
  \end{pmatrix} \,.
\]
Give the matrix which represents $F$ with respect to a basis of
your choice.

\end{enumerate}
\hfill(5+5+10+5+5)

%\newpage\mbox{}\newpage





\item
\begin{itemize}
\item[(E)] Consider the matrix
\[
  A =
  \begin{pmatrix}
    0 & 0 \\ 7 & 0    
    % Version B: 0 & 0 \\ 1 & 0    
    % Version A: 0 & 1 \\ 0 & 0
  \end{pmatrix} \,.
\]
Show that $\Ker A = \Range A$.
\hfill (8)
\item[(A)]
Let $A,B \in M(n \times n)$ with ${\Ker A} \cap {\Range B} = \{0\}$.
Show that $\dim \Ker AB = \dim \Ker B$.
\hfill (10)
\end{itemize}
%\newpage


\end{enumerate}


\end{document}
