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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Midterm I}
\date{March 5, 2003}
\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.




\begin{enumerate}

\item
\begin{enumerate}
\item Find the normal equation for the plane that passes through the
points $(0, -1, 1)$, $(2, 2, -1)$, and $(1, 1, 3)$.
\item Find the distance of the point $(1,1,1)$ to this plane.
\end{enumerate}
\hfill (5+5)



\item Find the general solution to the system of linear equations $A\x
= \b$ with
\[
  A =
  \begin{pmatrix}
  1& -2 &1& 1\\
  2 & -4 & 2 & -2 \\
  2 & -4 & 2 & 6 
  \end{pmatrix} \,,
  \qquad
  \b =
  \begin{pmatrix}
  1 \\ -2 \\ 6
  \end{pmatrix}
\]
Check your answer!
\hfill (10)

\item 
\begin{itemize}
\item[(E)] Consider the matrix $A$ from the previous question.  Show
that $A \x = \c$ does not have a solution when
\[
  \c =
  \begin{pmatrix}
  1 \\ 1 \\ -1
  \end{pmatrix} \,.
\] 
\hfill (4)

\item[(A)] Consider the matrix $A$ from the previous question.
Characterize all vectors $\c \in \R^3$ such that $A \x = \c$ is
solvable.
\hfill (5)

\end{itemize}


\item
\begin{itemize}
\item[(E)] Decide whether the following set of vectors is a basis of
$\R^3$.  Justify your answer.
\[
  \v_1 =
  \begin{pmatrix}
  1 \\ 2 \\ -1
  \end{pmatrix} \,, \quad
  \v_2 =
  \begin{pmatrix}
  2 \\ 3 \\ -1
  \end{pmatrix} \,, \quad
  \v_3 =
  \begin{pmatrix}
  -1 \\ 1 \\ -2
  \end{pmatrix} \,.
\]
\hfill(4)

\item[(A)] Prove that the columns of any $n\times k$ matrix with $n<k$
are linearly dependent.

\end{itemize} \hfill(5)


\item
\begin{itemize}
\item[(E)] Let $\displaystyle A = 
\begin{pmatrix}
  1 & 0 & 3 \\ 0 & 1 & 0 \\ 3 & 0 & 1
\end{pmatrix}\,.$
\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(8+4+4)

\item[(A)]
\begin{enumerate}
\item Find the matrix $A$ representing the linear transformation on
$\R^2$ that maps the triangle with vertices $\a=(0,0)$, $\b=(1,0)$,
$\c=(1,1)$ onto the triangle with vertices $\a'=(0,0)$,
$\b'=(\tfrac12,\tfrac32)$, $\c'=(2,2)$.

\item Diagonalize $A$.  I.e., find a diagonal matrix $D$ and a change
of coordinate $S$ such that $D=S^{-1}AS$.

\item Use the result from (b) to give a concise geometric description
of the linear transformation.
\end{enumerate}
\hfill(5+10+5)

\end{itemize}



\item Let $V$ be the vector space of continuous functions on $\R$
spanned by the basis
\[
  E = \{ e^{ix}, e^{-ix}\}
\]
with the usual addition and scalar multiplication.
\begin{enumerate}
\item Find the matrix $D$ which represents the derivative operator on
$V$ with respect to the basis $E$.
\item Let
\[
  T = \{ \sin x, \cos x \} \,.
\]
Explain why $T$ is also a basis of $V$, and find the matrix $S$ for
the change of basis from $T$ to $E$.

(Recall that $e^{ix}=\cos x + i \sin x$.)

\item Find the matrix $C$ which represents the derivative with
respect to the new basis $T$.
\end{enumerate}
\hfill(5+5+5)

\item \begin{itemize}
\item[(E)] If $A\in M(n\times n)$, show that $B=A^TA$ is a symmetric
matrix.

(Recall that a matrix $C$ is symmetric if $C=C^T$.)
\hfill (4)

\item[(A)] Prove that the eigenvalues of any real symmetric matrix are
real.
\hfill(5)

\end{itemize}


\end{enumerate}

\end{document}
