Given that $$\left[\begin{array}{cccc} -3 &1 &-3 &1\cr 3 &5 &0 &-1\cr -3 &2 &-1 &2 \end{array}\right] \sim \left[\begin{array}{cccc} 1 &0 &0 &-{\frac{2}{3}}\cr 0 &1 &0 &{\frac{1}{5}}\cr 0 &0 &1 &{\frac{2}{5}} \end{array}\right],$$ write $\left[\begin{array}{c} 1\cr -1\cr 2 \end{array}\right]$ as a linear combination of the vectors $\left[\begin{array}{c} -3\cr 3\cr -3 \end{array}\right], \left[\begin{array}{c} 1\cr 5\cr 2 \end{array}\right], \left[\begin{array}{c} -3\cr 0\cr -1 \end{array}\right].$

$\left[\begin{array}{c} 1\cr -1\cr 2 \end{array}\right] =$ $\left[\begin{array}{c} -3\cr 3\cr -3 \end{array}\right] +$ $\left[\begin{array}{c} 1\cr 5\cr 2 \end{array}\right] +$ $\left[\begin{array}{c} -3\cr 0\cr -1 \end{array}\right].$