\documentclass{article} 

\usepackage{amsmath,amsfonts}

\input{ExamMacros}

\prof{Peter Cholak}
%\prof{George McNinch} 
\class{Math 221} 
\quizdate{Wednesday, December 18, 1996.}
\examname{Final}


\begin{document}


%\pagestyle{empty}

\heading

\noindent  This final is worth 150 points. There are 7 problems on
5 pages and one blank page.  {\it Be sure to show all your work;
  you will not get full credit unless your work matches your
  answer!!}
\\

\begin{problem}
  (10 points) Let $A =
\begin{bmatrix}
  1 & 0 & 0 & 4 \\
  0 & 0 & 1 & 0 \\
  0 & -2 & 0 & 0 \\
  1/2 & 0 & 0 & 0 
\end{bmatrix}$. \\
If $b$ is some real number, use Cramer's rule to find the $x_3$
coordinate of the solution to the following equation:
$$A
\begin{bmatrix}
  x_1 \\ x_2 \\ x_3 \\ x_4
\end{bmatrix}
=
\begin{bmatrix}
  0 \\ b \\ -1 \\ -3 
\end{bmatrix}.$$
\end{problem} 

\newpage

\begin{problem}
  Let $A = \begin{bmatrix} 1&2&1&1 \\ 2&4&1&0 \\ 3&6&0&-3 \\
  \end{bmatrix}$. 
   \begin{subproblem}
     (10 points) Determine whether $A\vec{x} = \begin{bmatrix} 1
       \\ 0 \\ -3
     \end{bmatrix}$ is consistent and if so, find the general
     solution.
   \end{subproblem}
\newpage
   \begin{subproblem}
     (10 points) Determine whether $A\vec{x} = \begin{bmatrix} 1\\
       1 \\ 6\end{bmatrix}$ is consistent and if so, find the
     general solution.
   \end{subproblem}
  % \begin{subproblem}
%     Determine the value(s) of $b$ such that $A\vec{x} =
%     \begin{bmatrix} 1\\ -1 \\ b\end{bmatrix}$ is consistent and
%     find the general solution for this value(s).
%   \end{subproblem}
\end{problem}


\newpage %\vspace{5cm}

%\begin{problem}
%  Let $A = \begin{bmatrix} 1&1&-3&2 \\ 0&2&0&3 \\ -1&0&-3&-3 \\
%    0 & 0 & 0 & 2 \\ \end{bmatrix}$. Let $T_A (\vec{x}) = A
%  \vec{x}$. $T_A$ is a function from $\mathbb{R}^4$ to
%  $\mathbb{R}^4$. Is $T_A$ one-to-one? onto? Is $\vec{x} =
%    \begin{bmatrix} -5 \\ -2 \\ 1 \\ 0 \\ \end{bmatrix}$ an
%    eigenvector? If so, find the corresponding eigenvalue. Is
%    $\vec{y} = \begin{bmatrix} 0 \\ 2 \\ 0 \\ 2 \\
%    \end{bmatrix}$ an eigenvector? If so, find the corresponding
%    eigenvalue.
%\end{problem}

%\newpage
\begin{problem}
  (40 points -- 8 points each) For each of the following,
  indicate whether the statement is true or false.  Include a
  brief explanation of your answer.  {\it You will not receive
    full credit without a reasonable explanation!}
  \begin{subproblem}
    The rule $\langle \vec{x}, \vec{y} \rangle = x_1 y_1 - x_2
    y_2$ defines an inner product on the vector space
    $\mathbb{R}^2$.  (Here $\vec{x} = (x_1, x_2)$ and $\vec{y} =
    (y_1,y_2)$.)
  \end{subproblem}
%\vspace{6cm}
%  \begin{subproblem}
%    Let $A$ be a $3\times7$ matrix. Then the row vectors of $A$
%    {\em must} span $\mathbb{R}^3$.
%  \end{subproblem}
\vspace{6cm}
  \begin{subproblem}
    Let $W$ be all $2\times2$ matrices $\begin{bmatrix} a & b \\ c & d \\
    \end{bmatrix}$ such that $a+d=1$ and $V$ be all $2\times2$
    matrices. Then $W$ is a subspace of $V$.
  \end{subproblem}
  \vspace{8cm}
  \begin{subproblem}
    Let $A=  
    \begin{bmatrix}
      1 & 2 & 3 \\
      0 & -4 & 2 \\
      0 & 0 & b \\
    \end{bmatrix}$.
    Then for all values of $b$ the linear transformation
    $T_A(\vec{x}) = A \vec{x}$ is onto.
  \end{subproblem}
    \vspace{7cm}
  \begin{subproblem}
    Let $W$ be the set of $\vec{b}$ such that $A\vec{x}= \vec{b}$
    is consistent, where $A$ is an $4 \times 3$ matrix. Then the
    $W$ is a subspace of $\mathbb{R}^4$.
  \end{subproblem}
  \vspace{9cm}
  \begin{subproblem}
    Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear
    transformation. If $k_1 \vec{x}_1 + k_2 \vec{x}_2 = \vec{0}$
    then $k_1 T(\vec{x}_1) + k_2 T(\vec{x}_2) = \vec{0}$.
  \end{subproblem}
%\vspace{9cm}
 % \begin{subproblem}
%    Let $T(x,y,z) = ( x^2 +y^2, \sqrt{x+y}, z+1, 0)$ is a linear
%    transformation from $\mathbb{R}^3$ to $\mathbb{R}^4$.
%  \end{subproblem}
\end{problem} 


%\newpage
%\begin{problem}
%  (Here we are working in $\mathbb{R}^4$ with the Euclidean inner
%  product.)
%  \begin{subproblem}
%    Find an {\em orthonormal} basis for $W^\perp$, where $W =
%    \text{ span}\{ (1,1,0,0), (0,2,1,1) \}$ (you will need to use
%    Gram-Schmidt).
%  \end{subproblem}
%  \newpage
% \begin{subproblem}
%   Determine if $\vec{x}=(1,1,1,1)$ is in $W^\perp$. If so, write
%   $\vec{x}$ as a linear combination of basis elements. If not,
%   find $\text{Proj}_W(\vec{x})$.
% \end{subproblem}
%\end{problem}

\newpage %\vspace{7cm}
\begin{problem}
  (15 points) Consider the following three vectors in
  $\mathbb{R}^4$ (with the Euclidean inner product): $(1,0,0,1),
  (0,1,0,-1)$ and $(0,0,1,1)$. Find an {\em orthonormal} basis
  for the space spanned by these three vectors (you will need to
  use Gram-Schmidt).
\end{problem}


\newpage
\begin{problem}
  Consider the following system of linear equations:
$$  \begin{matrix}
    1 x_1 & + & 4 x_2 & + & 0 x_3 & = & a_1 \\[7pt]
    0 x_1 & + & 2 x_2 & + & 1 x_3 & = & a_1 \\[7pt]
    0 x_1 & + & 0 x_2 & + & 2 x_3 & = & a_1 \\[7pt]
  \end{matrix} $$

\begin{subproblem}
  (5 points) Find the coefficient matrix $P$ of this system of
  linear equations.
\end{subproblem}

\vspace{3cm}
\begin{subproblem}
  (10 points) Is $P$ invertible?  If so, find the inverse matrix
  $P^{-1}$. Otherwise, explain why $P$ has no inverse.
\end{subproblem}

\vspace{7cm}
\begin{subproblem}
  (5 points) Find the {\em general} solution $\vec{x} =
  \begin{bmatrix}
    x_1 \\ x_2 \\ x_3
  \end{bmatrix}$ 
  to this system of linear equations.
\end{subproblem}

\end{problem}

\newpage
\begin{problem}
  Let $A = 
  \begin{bmatrix}
    1 & 0 & 1 \\
    0 & 1 & 0 \\
    0 & 0 & -2 \\
  \end{bmatrix}$.

  \begin{subproblem}
    (15 points) Find the eigenvalues for the matrix $A$.  For
    each eigenvalue that you find, find a basis for the
    corresponding {\em eigenspace}.
  \end{subproblem}
  
\vspace{12cm}
  \begin{subproblem}
    (5 points) Find a matrix $P$ and a diagonal matrix $D$ such
    that $P^{-1} A P = D$.
  \end{subproblem}
 
\end{problem}

\newpage
\begin{problem}
  Consider the vectors $\vec{v} = \dfrac{1}{\sqrt{2}} (1,-1,0)$
  and $\vec{w} = (0,0,1)$.  These vectors form an {\em
    orthonormal} basis for a subspace $W$ in $\mathbb{R}^3$ (with
  the Euclidean inner product).  Let $P: \mathbb{R}^3 \to
  \mathbb{R}^3$ be the linear transformation given by
  orthogonally projecting onto the subspace $W$; so $P(\vec{x}) =
  \text{Proj}_W(\vec{x})$.

  \begin{subproblem}
    (10 points) We know that for a vector $\vec{x}$, $P(\vec{x})
    = \text{Proj}_W(\vec{x}) = (\vec{x} \cdot \vec{w})\vec{w} +
    (\vec{x} \cdot \vec{v})\vec{v}$.  Compute the standard matrix
    associated with $P$. (Hint: compute $P(\vec{e}_1)$,
    $P(\vec{e}_2)$ and $P(\vec{e}_3)$.)
  \end{subproblem}
   \vspace{6cm}

  (For the remaining parts of the problem there is nothing to
  compute.) 
  \begin{subproblem}
    (5 points) $\vec{z} = (1,1,0)$ is orthogonal to the subspace
    $W$.  Is the vector $\vec{z}$ an eigenvector for $P$?  If so,
    what is the eigenvalue?
  \end{subproblem}

\vspace{3cm}
  \begin{subproblem}
    (5 points)Is the vector $\vec{v}$ an eigenvector for $P$?  If
    so, what  is the eigenvalue. What about $\vec{w}$?
  \end{subproblem}
\vspace{3cm}
\begin{subproblem}
  5 points) Find the eigenvalues of the standard matrix for $P$.
  Find bases for the eigenspaces. (Hint: use the above two
  parts.)
\end{subproblem}
  
\end{problem}

\end{document} 

%%% Local Variables: 
%%% mode: latex 
%%% TeX-master: t 
%%% End:



