Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} 2 & 4 & 8 & 12 & 0 & -7 \\ -3 & -1 & -2 & 2 & -5 & 6 \\ -3 & -3 & -6 & -6 & -3 & -1 \\ 0 & 2 & 4 & 8 & -2 & -9 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & -2 & 2 & 0 \\ 0 & 1 & 2 & 4 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \]

1. The vector equation \( x_{1} \left[\begin{array}{c} 2 \\ -3 \\ -3 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -1 \\ -3 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 8 \\ -2 \\ -6 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} 12 \\ 2 \\ -6 \\ 8 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ -5 \\ -3 \\ -2 \end{array}\right] = \left[\begin{array}{c} -7 \\ 6 \\ -1 \\ -9 \end{array}\right] \)has no solutions.

2. \( \left[\begin{array}{c} -7 \\ 6 \\ -1 \\ -9 \end{array}\right] \) is not a linear combination of the vectors \( \left[\begin{array}{c} 2 \\ -3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ -2 \\ -6 \\ 4 \end{array}\right] , \left[\begin{array}{c} 12 \\ 2 \\ -6 \\ 8 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -5 \\ -3 \\ -2 \end{array}\right] \).