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\prof{Peter Cholak}
%\prof{George McNinch} 
\class{Math 221} 
\quizdate{Wednesday, December 4, 1996}
\examname{Exam 3}


\begin{document}


%\pagestyle{empty}

\heading

\noindent  This exam is worth 100 points. There are 5 problems on
6 pages and one blank page.  Be sure to show all your work; you
will not get full credit unless your work matches your answer!
\\

\begin{problem}
   Let $A = \begin{bmatrix} 1&2&0 \\ 1&4&1 \\ 1&0&-1 \end{bmatrix}$. 
   \begin{subproblem}
     (5 points) Find a basis for the row space of $A$.
   \end{subproblem}\vspace{.25cm}
   \begin{subproblem}
     (6 points) Find a basis for the column space of $A$.
   \end{subproblem}\vspace{.25cm}
   \begin{subproblem}
     (6 points) Find a basis for the nullspace of $A$.
   \end{subproblem}\vspace{.25cm}
   \begin{subproblem}
     (3 points) Determine the value of rank($A$) and
     nullity($A$).
    \end{subproblem}
\end{problem}
\newpage
\begin{problem}
  Let $A = \begin{bmatrix} 1&2&0 \\ 1&4&1 \\
    1&0&-1
  \end{bmatrix}$ (the same matrix as in the last problem). (Hint:
  consider the column space of $A$.)
   \begin{subproblem}
     (6 points) Determine whether $A\vec{x} = \begin{bmatrix} 1
       \\ 3 \\ -1 \end{bmatrix}$ is consistent and if so, find
     the general solution.
   \end{subproblem}
\vspace{9cm}
   \begin{subproblem}
     (6 points) Determine whether $A\vec{x} = \begin{bmatrix} 1\\
       0 \\ 4\end{bmatrix}$ is consistent and if so, find the
     general solution.
   \end{subproblem}\newpage
  % \begin{subproblem}
%     Determine whether there is a solution to $A\vec{x} =
%           \begin{bmatrix} 0\\ 0 \\ 0\end{bmatrix}$ and if so, find the 
%           general solution.
%   \end{subproblem}\vspace{8cm}
   \begin{subproblem}
     (8 points) Determine the value(s) of $b$ such that $A\vec{x}
     = \begin{bmatrix} 0\\ b \\ 2\end{bmatrix}$ is consistent and
     find the general solution for this value(s).
   \end{subproblem}
\end{problem}
\newpage
%\begin{problem}
%   Let a rectangular $x^{\prime}y^{\prime}z^{\prime}$-coordinate 
%   system be obtained by rotating a rectangular $xyz$-coordinate 
%   system counterclockwise about the $y$-axis (looking sideway at the 
%   $y$-axis) through the angle $\theta = \frac{\pi}{4}$.  Find the
%   $x^{\prime}y^{\prime}z^{\prime}$-coordinates of the point whose 
%   $xyz$-coordinates are $(w_{1},w_{2},w_{3})$.
%\end{problem}

\begin{problem}
  For each of the following, indicate whether the statement is
  true or false.  Include a brief explanation of your answer.
  (You will not full credit without a reasonable explanation!).


\begin{subproblem}
  (8 points) Let $V$ be an inner product space, and let $S = \{
  \vec{v}_1, \vec{v}_2 \}$ be an {\em orthogonal} set of vectors.
  Then $S$ is a linearly independent set of vectors.
\end{subproblem}
\vspace{7cm}
%\begin{subproblem}
%  The rule $\langle \vec{x}, \vec{y} \rangle = x_1 y_1 - x_2 y_2
%  + x_3 y_3$ defines an inner product on the vector space
%  $\mathbb{R}^3$.  (Here $\vec{x} = (x_1, x_2, x_3)$ and $\vec{y}
%  = (y_1,y_2,y_3)$.
%\end{subproblem}

%\begin{subproblem}
%  Let $A$ be a $7\times3$ matrix. Then the row vectors of $A$
%  {\em must} span $\mathbb{R}^3$.
%\end{subproblem}
%\vspace{4cm}
\begin{subproblem}
  (6 points) Let $A$ be a $3\times7$ then the general solution to
  the system of linear equations $A\vec{x} = \vec{b}$ has at
  least 4 parameters.
\end{subproblem}
\vspace{4cm}
\begin{subproblem}
  (6 points) Let S be a orthonormal basis for an inner product
  space $V$, $(\vec{u})_S = (1,2,0)$ and $(\vec{v})_S =
  (-1,2,1)$.  Then the distance between $\vec{u}$ and $\vec{v}$
  is 4, i.e.\ $d(\vec{u},\vec{v})= 4$.
\end{subproblem}
\end{problem} 
\newpage
%\begin{problem} In each of the following, please circle {\em each}
%  true statment.
%\begin{subproblem}
%  Let $A$ be a $5 \times 3$ matrix.  Suppose that the nullity of $A$
%  is 3.  Which of the following are true statements? 

%\qquad  \begin{minipage}[t]{4in}
% \begin{enumerate}
% \item The equations $A \vec{x} =
%    \begin{bmatrix}
%      1 \\ 0 \\ 0
%    \end{bmatrix}$ and $A \vec{x} =  
%    \begin{bmatrix}
%      0 \\ 1 \\ 0
%    \end{bmatrix}$
%    are both consistent.
%  \item $A$ has precisely two linearly independent columns.
%  \item $A$ has precisely two linearly independent rows.
%  \end{enumerate}
%\end{minipage}
%\end{subproblem}

%\begin{subproblem}
%  Let $M$ be a $2 \times 2$ matrix, let $\vec{r}_1$ and
%  $\vec{r}_2$ denote the rows of $M$, and assume that $\{
%  \vec{r}_1, \vec{r}_2 \}$ is an orthonormal basis of
%  $\mathbb{R}^2$. Which of the following are true?

%  \qquad
%  \begin{minipage}[t]{4in}
%    \begin{enumerate}
%    \item The columns of $M$ form an orthonormal basis for
%      $\mathbb{R}^2$.
%    \item The determinant of $M$ is 1.
%    \item $M$ is an invertible matrix.
%    \end{enumerate}
%  \end{minipage}
%\end{subproblem}
%\end{problem}


\begin{problem}
  (20 points) Use the Gram-Schmidt process to find an {\em
    orthogonal} basis for the row space of $A$, where
 % $A = 
%  \begin{bmatrix}
%    1 & 1 & 0  & 0\\
%    0 & 2 & 0  & 1 \\
%    2 & 0 & 1  & -4\\
%  \end{bmatrix}$.  
 $$A = 
  \begin{bmatrix}
    1 & 1 & 0  & 0\\
    0 & 2 & 0  & 1 \\
    1 & 3 & 0  & 1\\
  \end{bmatrix}$$

\end{problem}
\newpage
\begin{problem}
  Let $W$ denote the span of the following vectors in
  $\mathbb{R}^4$:
  
  $$\begin{matrix}
    \vec{b}_1 & = ~ ( & \frac{4}{5}, & 0, & \frac{3}{5}, & 0 & ) \\
    \vec{b}_2 & = ~ ( & -\frac{3}{5} , &  0 , & \frac{4}{5}, & 0 & ) \\
    \vec{b}_2 & = ~ ( & 0, & 1, & 0, & 1 & )
  \end{matrix}$$

  \begin{subproblem}
    (4 points) Verify that $\{ \vec{b}_1, \vec{b}_2, \vec{b}_3
    \}$ is an orthogonal basis for $W$.
  \end{subproblem}
\vspace{4cm}
  \begin{subproblem}
    \label{projection}
    (9 points) Let $\vec{x} = (1,2,1,0)$.  Compute
    $\text{Proj}_W(\vec{x})$, the {\em orthogonal projection} of
    $\vec{x}$ onto the subspace $W$.
  \end{subproblem}
\vspace{8cm}
  \begin{subproblem}
    (7 points) Find an orthonormal basis for $W^\perp$.  (Hint:
    first think about what the dimension of $W^\perp$ should be
    and then think about what you did in the last part.)
  \end{subproblem}
\end{problem}

\end{document}


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