To apply Theorem BS, we formulate a matrix $A$ whose columns are $\vect{v}_1,\,\vect{v}_2,\,\vect{v}_3,\,\vect{v}_4,\,\vect{v}_5$. Then we row-reduce $A$. After row-reducing, we obtain
\begin{equation*}
\begin{bmatrix}
\leading{1} & 0 & 0 & 2 & -1\\
0 & \leading{1} & 0 & 1 & -2\\
0 & 0 & \leading{1} & 0 & 0
\end{bmatrix}
\end{equation*}
From this we see that the pivot columns are $D=\set{1,\,2,\,3}$. Thus
\begin{equation*}
T=\set{\vect{v}_1,\,\vect{v}_2,\,\vect{v}_3}=\set{\colvector{2\\1\\1},\,\colvector{-1\\-1\\1},\,\colvector{1\\2\\3}}
\end{equation*}
is a linearly independent set and $\spn{T}=W$. Compare this problem with Exercise LI.M50.