Answer
A basis for the subspace is given by
$$\{2,5,-3,-2 ),(0,2,-1,-7),(0,0,-1,19) \}.$$
Work Step by Step
Let $S$ be given by
$$
S=\{(2,5,-3,-2 ),(-2,-3,2,-5),(1,3,-2,2) (-1,-5,3,5) \} .
$$
We form the matrix
$$ \left[ \begin {array}{cccc} 2&5&-3&-2\\ -2&-3&2&-5
\\ 1&3&-2&2\\ -1&-5&3&5
\end {array} \right]
.
$$
The reduced form of the matrix is given by
$$\left[ \begin {array}{cccc} 2&5&-3&-2\\ 0&2&-1&-7
\\ 0&0&-1&19\\ 0&0&0
&0\end {array} \right]
.$$
A basis for the subspace is given by
$$\{2,5,-3,-2 ),(0,2,-1,-7),(0,0,-1,19) \}.$$
