(1 point) Find a set of vectors $\{\vec{u}, \vec{v}\}$ in $\mathbb{R}^{4}$ that spans the solution set of the equations
\[
\left.\begin{array}{c}
\left\{\begin{array}{r}
w-x-y-3 z=0 \\
w+2 x-y+2 z=0
\end{array}\right. \\
\vec{u}=\left[\begin{array}{l}
\square \\
\square
\end{array}\right] \\
\square
\end{array}\right]
\]
(The components of these vectors appear in alphabetical order: $(w, x, y, z))$.

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 \(\boxed{\left\{\begin{bmatrix} 1 \ 0 \ 1 \ 0 \end{bmatrix}, \begin{bmatrix} \frac{1}{3} \ -\frac{5}{3} \ 0 \ 1 \end{bmatrix}\right\}}\)