Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} 1 & 2 & 2 & -12 & 0 & 7 \\ -2 & 1 & 2 & -4 & 7 & 2 \\ 1 & -4 & -5 & 21 & -8 & -5 \\ -1 & 2 & -2 & 4 & -4 & -3 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & -2 & -2 & 0 \\ 0 & 1 & 0 & -2 & -1 & 0 \\ 0 & 0 & 1 & -3 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

1. The vector equation \( x_{1} \left[\begin{array}{c} 1 \\ -2 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 2 \\ -5 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -12 \\ -4 \\ 21 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 7 \\ -8 \\ -4 \end{array}\right] = \left[\begin{array}{c} 7 \\ 2 \\ -5 \\ -3 \end{array}\right] \)has a solution.

2. \( \left[\begin{array}{c} 7 \\ 2 \\ -5 \\ -3 \end{array}\right] \) is not a linear combination of the vectors \( \left[\begin{array}{c} 1 \\ -2 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -5 \\ -2 \end{array}\right] , \left[\begin{array}{c} -12 \\ -4 \\ 21 \\ 4 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 7 \\ -8 \\ -4 \end{array}\right] \).