Answer:

\[\operatorname{RREF} \left[\begin{array}{ccccc|c} -1 & 0 & 3 & -7 & -6 & 1 \\ -1 & -2 & 2 & -9 & 2 & 3 \\ 4 & 2 & 4 & 0 & -14 & 6 \\ 2 & -1 & 2 & -4 & -1 & -10 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 2 & -3 & 0 \\ 0 & 0 & 1 & -2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

1. The vector equation \( x_{1} \left[\begin{array}{c} -1 \\ -1 \\ 4 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -2 \\ 2 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 2 \\ 4 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} -7 \\ -9 \\ 0 \\ -4 \end{array}\right] + x_{5} \left[\begin{array}{c} -6 \\ 2 \\ -14 \\ -1 \end{array}\right] = \left[\begin{array}{c} 1 \\ 3 \\ 6 \\ -10 \end{array}\right] \)has no solutions.

2. \( \left[\begin{array}{c} 1 \\ 3 \\ 6 \\ -10 \end{array}\right] \) is not a linear combination of the vectors \( \left[\begin{array}{c} -1 \\ -1 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -7 \\ -9 \\ 0 \\ -4 \end{array}\right] , \text{ and } \left[\begin{array}{c} -6 \\ 2 \\ -14 \\ -1 \end{array}\right] \).