Step 1 :Given the system of equations: \(w-x-y-3z=0\) and \(w+2x-y+2z=0\)
Step 2 :We can solve this system of equations to find the solution set in terms of \(w\), \(x\), \(y\), and \(z\)
Step 3 :By solving, we find that the solution set can be expressed as a linear combination of the vectors \(\vec{u} = \begin{bmatrix} 1 \ 0 \ 1 \ 0 \end{bmatrix}\) and \(\vec{v} = \begin{bmatrix} \frac{1}{3} \ -\frac{5}{3} \ 0 \ 1 \end{bmatrix}\)
Step 4 :Thus, the set of vectors that spans the solution set is \(\boxed{\left\{\begin{bmatrix} 1 \ 0 \ 1 \ 0 \end{bmatrix}, \begin{bmatrix} \frac{1}{3} \ -\frac{5}{3} \ 0 \ 1 \end{bmatrix}\right\}}\)