Answer:

\[\operatorname{RREF} \left[\begin{array}{cccc|c} 4 & -4 & -1 & -1 & -4 \\ 0 & 0 & 0 & 0 & 8 \\ 2 & 2 & 4 & 0 & -3 \\ -1 & 2 & 2 & 1 & 4 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right] \]

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

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