Finding the basis of the intersection of a subspace and span

The next step is to note that\begin{align}U_1\cap U_2&=\left\{\begin{pmatrix}\lambda\\\lambda-\mu\\0\\3\lambda+\mu\end{pmatrix}\,\middle|\,\lambda-(\lambda-\mu)-3(3\lambda+\mu)=0\right\}\\&=\left\{\begin{pmatrix}\lambda\\\lambda-\mu\\0\\3\lambda+\mu\end{pmatrix}\,\middle|\,9\lambda+2\mu=0\right\}\\&=\left\{\begin{pmatrix}\lambda\\\frac{11}2\lambda\\0\\-\frac32\lambda\end{pmatrix}\,\middle|\,\lambda\in\mathbb{R}\right\}.\end{align}Can you take it from here?

$U_1$ is the set of vector such that $\begin{bmatrix} 1 &-1 &1 &-3\end{bmatrix}\begin{bmatrix} x_1 \\x_2 \\x_3 \\x_4\end{bmatrix} = 0$

$\begin{bmatrix} 1 &-1 &1 &-3\end{bmatrix}\begin{bmatrix} 1 \\1 \\0 \\3\end{bmatrix} = -9\\ \begin{bmatrix} 1 &-1 &1 &-3\end{bmatrix}\begin{bmatrix} 0 \\-1 \\0 \\1\end{bmatrix} = -2$

$2\begin{bmatrix} 1 \\1 \\0 \\3\end{bmatrix} - 9\begin{bmatrix} 0 \\-1 \\0 \\1\end{bmatrix} = \begin{bmatrix} 2 \\11 \\0 \\-3\end{bmatrix}$