Consider the subspace $W=\spn{\set{\vect{v}_1,\,\vect{v}_2,\,\vect{v}_3,\,\vect{v}_4}}$. Find a set $S$ so that (1) $S$ is a subset of $W$, (2) $S$ is linearly independent, and (3) $W=\spn{S}$. Write each vector not included in $S$ as a linear combination of the vectors that are in $S$. \begin{align*} \vect{v}_1&=\colvector{1\\-1\\2} & \vect{v}_2&=\colvector{4\\-4\\8} & \vect{v}_3&=\colvector{-3\\2\\-7} & \vect{v}_4&=\colvector{2\\1\\7} \end{align*}