Answer
A basis for the subspace is given by
$$\{2,9,-2,53),(0,31,0,155),(0,0,0,1)\}.$$
These answers are wrong and they don't even show the work
Work Step by Step
Let $S$ be given by
$$
S=\{ (2,9,-2,53),(-3,2,3,-2),(8,-3,-8,17)(0,-3,0,15) \}.
$$
We form the matrix
$$ \left[ \begin {array}{ccccc} 2&9&-2&53\\ -3&2&3&-2\\ 8&-3&-8&17\\0&-3&0&15
\end {array} \right].
$$
The reduced form of the matrix is given by
$$\left[ \begin {array}{cccc} 2&9&-2&53\\ 0&31&0&155\\ 0&0&0&30
\\ 0&0&0&0\end {array} \right]
.$$
A basis for the subspace is given by
$$\{2,9,-2,53),(0,31,0,155),(0,0,0,1)\}.$$
