Answer
A basis for the subspace is given by
$$\{(4,4,8),(0,0,1)\}.$$
Work Step by Step
Let $S$ be given by
$$
S=\{(4,4,8),(1,1,2),(1,1,1)\}
$$
We form the matrix
$$ \left[ \begin {array}{ccccc} 4&4&1\\ 1&1&2\\ 1&1&1
\end {array} \right].
$$
The reduced form of the matrix is given by
$$\left[ \begin {array}{ccc} 4&4&8\\ 0&0&1
\\ 0&0&0\end {array} \right].$$
A basis for the subspace is given by
$$\{(4,4,8),(0,0,1)\}.$$
