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\newcommand{\courseblurb}{Linear Algebra}

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\begin{document}

\doctitle

\begin{problem} (10 points)
  Consider the following system of linear equations:
\begin{center}
  $\begin{matrix}
    2x_1 & + & 4x_2 & + & 6x_3 & = & 6 \\
    x_1 & + & 3x_2 & + & 4x_3 & = & 4 \\
    &     & x_2 & + & ax_3 & = & 1 \\
\end{matrix}$
\end{center}

\begin{subproblem} (1 point)
  Find $A$ and $\vec{b}$ such that the above system is in the form $A
  \vec{x} = \vec{b}$ (i.e.\ an matrix equation).
\end{subproblem}


\begin{subproblem} (5 points)
  Explicitly showing \emph{all} the steps reduce the \emph{augmented
    matrix} into reduce echelon form.  (Hint: there are 2 cases for
  the rest of the problem.)
\end{subproblem}



\begin{subproblem} (1 point)
  What is the rank of $A$? 
\end{subproblem}



\begin{subproblem} (1 point)
  Is there an $a$ such that the system does not have a solution?  Why
  or why not?
\end{subproblem}         



\begin{subproblem} (2 points)
  Find the general solution (in vector form) for the above system.
\end{subproblem}

%\begin{subproblem} (4 points)
%  Find a value for $a$ and a vector $\vec{c}$ such that $A\vec{x} =
%  \vec{c}$ is inconsistent.  
%\end{subproblem}
\end{problem}

\begin{problem} (6 points)
  Let $\vec{x}_1 = \begin{bmatrix} 1 \\ -2 \\ 4 \end{bmatrix}$,
  $\vec{x}_2 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$ and $
  \vec{y} = \begin{bmatrix} 4 \\0 \\ b \end{bmatrix}$.

  \begin{subproblem} 
    Find a value of $b$ such that $\vec{y}$ \emph{is} a linear
    combination of $\vec{x}_1$ and $\vec{x_2}$.
  \end{subproblem}


\begin{subproblem} 
  Find a value of $b$ such that $\vec{y}$ \emph{is not} a linear
  combination of $\vec{x}_1$ and $\vec{x_2}$.
  \end{subproblem}
\end{problem}



\begin{problem} ( 4 points)
  Both of the following statements are false.  Provide an example.

  \begin{subproblem} (2 points)
    If $A$ is a $5 \times 4$ matrix and has rank $4$ then for all
    $\vec{b} \in \mathbb{R}^5$, $A\vec{x} = \vec{b}$ is consistent.
  \end{subproblem}
 



 \begin{subproblem} (2 points)
    If $A$ is an upper triangular matrix then the rank of $A$ is the
    number of non-zero entries on $A$'s diagonal.  
  \end{subproblem}



\end{problem}



\end{document}

