Show how to find the solution set for the following vector equation
\[ x_{1} \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 1 \\ -1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 2 \\ -2 \\ 8 \\ 3 \end{array}\right] .\]
Answer:
\[\mathrm{RREF} \left[\begin{array}{ccc|c} 3 & -1 & 1 & 2 \\ 1 & 0 & 1 & -2 \\ 2 & -2 & -1 & 8 \\ 1 & 0 & 0 & 3 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \]
The solution set is \( \left\{\right\} \).