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\prof{Peter Cholak}
%\prof{George McNinch} 
\class{Math 221} 
\quizdate{Friday, November 1}
\examname{Exam 2}


\begin{document}


%\pagestyle{empty}

\heading

\noindent  This exam is worth 100 points. There are 7 questions
on 3 pages (both sides) and one blank page.

\bigskip

\begin{problem}
  (12 points) Let $b$ be a fixed real number. Use {\em Cramer's
    Rule} to find the $y$-coordinate of the solution to the
  following system of linear equations.  No credit will be given
  for the use of methods other than Cramer's Rule.
\begin{displaymath}
  \label{eq1}
  \begin{bmatrix}
    -3 & 0 & 2 \\
    0 & 1 & 1 \\
    0 & 0 & 3
  \end{bmatrix}
  \cdot
  \begin{bmatrix}
    x \\ y \\ z
  \end{bmatrix}
  =
  \begin{bmatrix}
    1 \\ 0 \\ b
  \end{bmatrix}
\end{displaymath}
\end{problem}

\vspace{8cm}

\begin{problem}
  (13 points) Find a basis and the dimension of the {\em solution
    space} to the equation
  \begin{displaymath}
    \begin{bmatrix}
      3 & 2 & -2 \\
    \end{bmatrix}
    \begin{bmatrix}
      x \\ y \\ z
    \end{bmatrix}
    = 0.
  \end{displaymath}
  
\end{problem}

\newpage

\begin{problem}
  Let $T : \mathbb{R}^2 \to \mathbb{R}^2$ be the linear
  transformation which first reflects a vector about the
  $x$-axis, and then rotates the vector counter-clockwise by
  $\frac{3\pi}{4}$ radians ($135$ degrees).

  \begin{subproblem}
    (9 points) Compute $T(\vec{e}_1)$ and $T(\vec{e}_2)$.
  \end{subproblem}
  \vspace{6cm}

  \begin{subproblem}
    (6 points) Determine the {\em standard matrix} $[T]$ of the
    transformation $T$.
  \end{subproblem}
  \vspace{6cm}

  \begin{subproblem}
    (5 points -- no partial credit) Decide which of the following
    accurately describes the effect of the linear transformation
    $T^{-1}$ on a vector in $\mathbb{R}^2$.

   \begin{enumerate}
   \item $T$ does not have an inverse.
     
   \item $T^{-1}$ first reflects the vector about the $x$-axis,
     and then it rotates the vector counter-clockwise
     $\frac{3\pi}{4}$ radians.
     
   \item $T^{-1}$ first reflects the vector about the $x$-axis, and
     then it rotates the vector clockwise $\frac{3\pi}{4}$ radians.
    
   \item $T^{-1}$ first rotates the vector counter-clockwise
     $\frac{3\pi}{4}$ radians, and then reflects the vector about the
     $x$-axis.
    
   \item $T^{-1}$ first rotates the vector clockwise $\frac{3\pi}{4}$
     radians, and then reflects the vector about the $x$-axis.
  \end{enumerate}

  \end{subproblem}
\end{problem}

\newpage

\begin{problem}
  Let $S:\mathbb{R}^3 \to \mathbb{R}^3$ be the linear
  transformation given by the rule $S(x,y,z) = (2x+y-4z,x+y,3z)$.

  \begin{subproblem}
    (6 points) Is $(0,\pi,0)$ an eigenvector for $S$?  If so, what
    is the corresponding eigenvalue?
  \end{subproblem}
  \vspace{8cm}
%  \begin{subproblem}
%    Is $(0,0,1)$ eigenvector for $S$?  If so, what is the
%    corresponding eigenvalue?
%  \end{subproblem}\vspace{6cm}
  \begin{subproblem}
    (10 points) Let $\vec{v} = (2d,d,1)$, $d$ a real number.  Is
    there a value $d$ such that $\vec{v}$ is an eigenvector for $S$?
    If so, what is the corresponding eigenvalue.
  \end{subproblem}
\end{problem}

\newpage
%\vspace{4cm}
\begin{problem}
  In the following, we specify a vector space $V$ and a subset
  $W$ of $V$.  In each case, determine if $W$ is a {\em subspace}
  of $V$.  Justify your conclusion.

  \begin{subproblem}
    (5 points) $V = \text{Mat}_{2 \times 2}$, the vector space of
    two by two matrices. \\
    
    $W =$ all two by two matrices $A$ with the property that $A
    \begin{bmatrix}
      -3 \\ 2
    \end{bmatrix}
    =
    \begin{bmatrix}
      0 \\ 0
    \end{bmatrix}
    $.
  \end{subproblem}
  \vspace{8cm}


%  \begin{subproblem}
%    $V = P_3$, the vector space of all polynomials of degree $\le 3$. \\
      
%    $W =$ all polynomials $f$ in $P_3$ so that $f(-1) = 0$.
%  \end{subproblem}
  
  
  \begin{subproblem}
    (5 points) $V = \mathbb{R}^3$. \\
      
    $W =$ all vectors $\vec{x} = (x_1,x_2,x_3)$ so that $\vec{x}
    \cdot (0,3,-1) = 5$.
  \end{subproblem}
\end{problem}


\newpage
\begin{problem}
  Consider the subset $B$ of $\mathbb{R}^4$ consisting of the
  following three vectors:
  $$
  \vec{b}_1 = (2,0,2,0) \quad \vec{b}_2 = (0,1,0,1) \quad
  \vec{b}_3 = (0,1,0,-1)$$

\begin{subproblem}
  (10 points) Is $B$ a basis for $\text{Span}(B)$? Why or why
  not?
\end{subproblem}
\vspace{10cm}

\begin{subproblem}
  (11 points) Decide if $\vec{v} = (1,-1,-1,2)$ is in
  $\text{Span}(B)$.  If so, write $\vec{v}$ as a linear
  combination of $b_1,b_2,b_3$. If not, is
  $\{\vec{u},\vec{b}_1,\vec{b}_2,\vec{b}_3\}$ a basis for
  $\mathbb{R}^4$?
\end{subproblem}  
\end{problem}
 
\newpage
\begin{problem}
  Decide whether the following assertion is true or
  false.  Explain your decision.

 % \begin{subproblem}
%    The set $S = \{(1,0,0,0),(0,1,1,0),(0,2,2,0) \}$ can be extended to a 
%    basis of $\mathbb{R}^4$.
%  \end{subproblem}
%\vspace{2.5cm}

%  \begin{subproblem}
    (8 points) Let $\vec{v}$ be a vector in $\mathbb{R}^3$.  If
    $\vec{v}$ is orthogonal to $\vec{e}_2$, then $\vec{v}$ is in
    $\text{Span}(\vec{e}_1,\vec{e}_3)$.
%  \end{subproblem}
  \vspace{8cm}

% \begin{subproblem}
%    (4 points) There are 5 vectors which span $\text{Mat}_{3
%      \times 2}$, the vector space of $3\times2$ matrices.
%  \end{subproblem}
%  \vspace{2.5cm}
\end{problem}



\end{document}


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