\begin{exercise}{E1}{Linear systems, vector equations, and augmented matrices}{0003}
\begin{exerciseStatement}
Consider the augmented matrix
\[ \left[\begin{array}{ccc|c}
1 & -1 & -1 & -6 \\
0 & 1 & -3 & 4 \\
0 & 0 & 0 & 1 \\
0 & 2 & -6 & 7
\end{array}\right] \]
\begin{enumerate}[(a)]
\item Write a system of scalar equations corresponding to this augmented matrix.
\item Write a vector equation corresponding to this augmented matrix.
\end{enumerate}
\end{exerciseStatement}
\begin{exerciseAnswer}
\begin{enumerate}[(a)]
\item \[\begin{matrix}
x_{1} & - & x_{2} & - & x_{3} & = & -6 \\
& & x_{2} & - & 3 \, x_{3} & = & 4 \\
& & & & 0 & = & 1 \\
& & 2 \, x_{2} & - & 6 \, x_{3} & = & 7 \\
\end{matrix}\]
\item \[ x_{1} \left[\begin{array}{c}
1 \\
0 \\
0 \\
0
\end{array}\right] + x_{2} \left[\begin{array}{c}
-1 \\
1 \\
0 \\
2
\end{array}\right] + x_{3} \left[\begin{array}{c}
-1 \\
-3 \\
0 \\
-6
\end{array}\right] = \left[\begin{array}{c}
-6 \\
4 \\
1 \\
7
\end{array}\right] \]
\end{enumerate}
\end{exerciseAnswer}
\end{exercise}
