Chapter 5.SE, Problem 10SE

In Exercises 7-11, use the fact that the matrix

$\left[A|b\right]=\left[\begin{array}{cccccc}1& -1& 3& 1& 3& 2\\ 1& 0& 2& 3& 2& 3\\ 0& -2& 2& -4& 3& 0\\ 2& -1& 5& 4& 6& 7\end{array}|\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]$

is row equivalent to

$\left[\begin{array}{cccccc}1& 0& 2& 3& 0& -1\\ 0& 1& -1& 2& 0& 3\\ 0& 0& 0& 0& 1& 2\\ 0& 0& 0& 0& 0& 0\end{array}|\begin{array}{c}4a-3b-2c\\ -3a+3b+c\\ -2a+2b+c\\ a-3b-c+d\end{array}\right]$.

If $S=\left\{\left[\begin{array}{cc}1& 1\\ 0& 2\end{array}\right],\left[\begin{array}{cc}-1& 0\\ -2& -1\end{array}\right],\left[\begin{array}{cc}3& 2\\ 2& 5\end{array}\right],\left[\begin{array}{cc}1& 3\\ -4& 4\end{array}\right],\left[\begin{array}{cc}3& 2\\ 3& 6\end{array}\right],\left[\begin{array}{cc}2& 3\\ 0& 7\end{array}\right]\right\}$,

then give an algebraic specification for Sp(S) and use the specification to determine a basis for Sp(S).