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\begin{center}
{\Large TEST 1

Math 315}
\end{center}

\begin{itemize}
\item  Show all your work and make your work neat.  You will be graded on this!
\item  You will be graded on both your answer and the work done to get that answer.
\item  You can use your calculator, but I should be able to follow your work as if you had no calculator!
\item  There are four (4) problems, each worth 10 points.
\item  Do your work on other paper.
\end{itemize}


\begin{enumerate}

\item  Consider the system of equations:
\[
\begin{array}{rrrrrrrrrrr}
x_1 & + & 2x_2 &  &   & + & 4x_4 & - & 5x_5 & = & 2 \\
    &   &      &   & x_3 & - & 2x_4 & + & 4x_5 & = & -7
\end{array}
\]
\begin{enumerate}
\item  As we know, this system of equations can also be written as a matrix equation $AX=B$.
  What are the matrices $A$, $X$ and $B$?
  \label{what is X}
\item  Write the augmented matrix associated to the system of equations.
\item  Reduce the augmented matrix to reduced row echelon form.
\item  Find the general solution to this system of equations.
  Write your solution as a vector (as in the vector $X$ of part~(\ref{what is X}))
  and in ``parametric form.''
\end{enumerate}

\item  
  \begin{enumerate}
  \item  What is a homogeneous system of equations?
    What is a non-homogeneous system of equations?
    \label{homo}
    Phrase your answers in terms of matrices (see problem~~(\ref{what is X})).
  \item  What does it mean for a system to be inconsistent?
    What does it mean for a system to be consistent?
  \item  Prove that a homogeneous system is always consistent.
  \item  Translate the fact that a homogeneous system is always consistent into a statement about the 
    null space of a matrix (this relates to part~(\ref{homo})).
  \end{enumerate}

\newpage

\item  Which of the following matrices are in reduced row echelon form?
  For the matrices \emph{not} in reduced row echelon form, perform row operations 
  to put the matrix in reduced row echelon form.
  \\
  (I must be able to see your work!  Doing it on your calculator is not enough!)
  \[
  A = \left[
    \begin{array}{rrrr}
      1 & 0 & 2 & 1 \\
      0 & 1 & -4 & 0 \\
      0 & 0 & 0 & 1 
    \end{array}
    \right]
  \quad
  B = \left[
    \begin{array}{rrrr}
      0 & 1 & 2 & 1 \\
      1 & 0 & -4 & 0 \\
      0 & 0 & 0 & 0 
    \end{array}
    \right]
  \quad
  C = \left[
    \begin{array}{rrrr}
      -1 & 0 & 2 & 0 \\
      0 & 2 & -4 & 0 \\
      0 & 0 & 0 & 1 
    \end{array}
    \right]
  \]



\item  For this problem, you must use the following (you don't have to do the reduction!):
  \\
  Suppose the following matrix is put into reduced row echelon form as follows:
  %
  \begin{equation} \label{row reduce}
  A = \left[ 
    \begin{array}{rrrrr}
      1 & 2 & -1 & 0 & -1 \\
      2 & 4 &  3 &  1 &  2 \\
      3 & 6 & 2 & 1 & 1 
    \end{array}
    \right]
  \quad
  \stackrel{\mbox{rref}}{\longrightarrow}
  \quad
  B =
  \left[
    \begin{array}{rrrrr}
      1 & 2 & 0 & \frac{1}{5} & -\frac{1}{5} \\
      0 & 0 & 1 &  \frac{1}{5} &  \frac{4}{5} \\
      0 & 0 & 0 & 0 & 0 
    \end{array}
    \right]
  \end{equation}
  %
  (So, if we perform row reduction on matrix $A$, we arrive at matrix $B$).
  \\
  Consider the set of vectors:
  \[
  S = \{ v_1, v_2, v_3, v_4, v_5 \}
  \]
  where 
  \[
  v_1 = \left[
    \begin{array}{r}
      1 \\ 2 \\ 3
    \end{array}
      \right]
  \quad
  v_2 = \left[
    \begin{array}{r}
      2 \\ 4 \\ 6
    \end{array}
      \right]
  \quad
  v_3 = \left[
    \begin{array}{r}
      -1 \\ 3 \\ 2
    \end{array}
      \right]
  \quad
  v_4 = \left[
    \begin{array}{r}
      0 \\ 1 \\ 1
    \end{array}
      \right]
  \quad
  v_5 = \left[
    \begin{array}{r}
      -1 \\ 2 \\ 1
    \end{array}
      \right]
  \]
  \begin{enumerate}
  \item  The row reduction  above, (\ref{row reduce}), is actually solving equations
    that will determine if the set $S$ is linearly dependent or linearly independent.
    What is the equation that we are solving?
    Explain why your equation yields the matrix $A$ above.
    (The equation should be written in terms of the vectors $v_1, v_2, v_3, v_4, v_5$.)
  \item  Using the matrices given above, find a maximal linear independent subset $M$, of $S$.
    Give some justification as to why your set is linearly independent and why it is a
    maximal linearly independent set.
  \item  Write the vectors \emph{not} in $M$ (your maximial linear independent subset of $S$)
    as linear combinations of vectors in $M$.
  \item  True or false, justify your answer:
    \[
    \mathrm{Span}\{v_1, v_2, v_3 \} = \mathrm{Span}\{v_1, v_3, v_4, v_5\}
    \]
  \end{enumerate}

\end{enumerate}

\end{document}