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