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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Midterm I}
\date{March 3, 2004}
\maketitle

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%% \vspace*{\fill}
%% \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{itemize}
\item[(E)] Find the distance of the point $\boldsymbol{p}=(0,1,0)^T$ to
  the plane
\[
  (\x-\a) \cdot \n = 0
\]
where $\a=(1,0,1)^T$ and $\n=\dfrac1{\sqrt3}\,(1, -1, -1)^T$.

\hfill (8)

\item[(A)] Show that two lines in $\R^3$ lie in a plane only if either
  they intersect or they are parallel.
\hfill (10)

\end{itemize}


\item Find the general solution to the system of linear equations $A\x
= \b$ with
\[
  A =
  \begin{pmatrix}
  0&  1 &2& 3\\
  -1& -2&-3 & -4 \\
  1 & 1& 1 & 1 
  \end{pmatrix} \,,
  \qquad
  \b =
  \begin{pmatrix}
  6 \\ -10\\ 4
  \end{pmatrix}
\]
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
\[
  \c =
  \begin{pmatrix}
  1 \\ 0 \\  0
  \end{pmatrix} \,.
\] 
\hfill (8)

\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 (10)

\end{itemize}


\item Let $\displaystyle A = 
\begin{pmatrix}
  1 & i \\ -i & 1
\end{pmatrix}\,.$
\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(10+5+5)



\item
\begin{itemize}
\item[(E)] Give an example of two matrices $A$ and $B$ such that $AB
  \neq BA$.  (An explicit calculation is required!)
\hfill(8)

\item[(A)] Let $A,B \in M(n\times n)$ such that there exists a basis
  $\{\v_1, \dots, \v_n \}$ of $\R^n$ that is also a set of
  eigenvectors for both $A$ and $B$.  Show that, in this case,
  $AB=BA$. 
\end{itemize}
\hfill(10)


\item
\begin{itemize}
\item[(E)] Let
\[
  \v_1 = 
  \begin{pmatrix}
  2 \\-2 \\-1
  \end{pmatrix} \,, \quad
  \v_2 = 
  \begin{pmatrix}
  2 \\1 \\-2
  \end{pmatrix} \,, \quad
  \v_3 = 
  \begin{pmatrix}
  3 \\-1 \\-2
  \end{pmatrix} \,.
\]

\begin{enumerate}
\item Let the linear transformation $F$ be the reflection about the
plane spanned by the standard unit vectors $\e_1$ and $\e_2$.  Find
the matrix representing $F$ in the standard basis.

\item Check that $\{\v_1, \v_2, \v_3\}$ form a basis of $\R^3$.

\item Find the matrix representing $F$ in the basis $\{\v_1, \v_2,
\v_3\}$.

\item Let $\a = 4\v_1-2\v_2+\v_3$; compute $F(\a)$ in the basis
$\{\v_1, \v_2, \v_3\}$.
\end{enumerate}

\hfill(5+5+9+5)


\item[(A)] Let $P_2$ be the vector space of polynomials of degree less
  or equal than $2$.
\begin{enumerate}
\item Show that
\[
  F(p) = 
  \begin{pmatrix}
    p(0) \\ p(1) \\ p(2)
  \end{pmatrix}
\]
is a linear transformation from $P_2$ into $\R^3$.

\item Find the matrix representing $F$ when $P_2$ is endowed with
  basis $B=\{1,x,x^2\}$, and $\R^3$ is endowed with its standard
  basis.

\item Find the matrix representing
\[
  G(p) = 
  \begin{pmatrix}
    p'(0) \\ p'(1) \\ p'(2)
  \end{pmatrix}
\]
with respect to the same pair of bases.

\item Is $F$ invertible?  Is $G$ invertible?  Explain.

\item Find a matrix $A$ such that
\[
  A
  \begin{pmatrix}
    p(0) \\ p(1) \\ p(2)
  \end{pmatrix}
  =
  \begin{pmatrix}
    p'(0) \\ p'(1) \\ p'(2)
  \end{pmatrix}
\]
for every $p\in P_2$.

\end{enumerate}

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


\end{document}
