$U,W$ are sub-spaces of $M^\mathbb{R}_{2x2}$ $$U=Sp\left\{\begin{pmatrix} 1 & 2\\ 4 & 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & -1\\ 3 & 2 \\ \end{pmatrix}, \begin{pmatrix} 1 & 11\\ 7 & -2 \\ \end{pmatrix} \right\}$$ $$W=Sp\left\{\begin{pmatrix} 1 &5\\ 5 & 0 \\ \end{pmatrix}, \begin{pmatrix} 0 & 1\\ 1 & 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & 1\\ 1 & -4 \\ \end{pmatrix} \right\}$$
Find a base for $U \cap W$
I got a bit stuck on this question not sure on how to use Gaussian elimination in order to find the base
HINT:
$M^\mathbb{R}_{2x2}$ is $\Bbb R^4$ written a little bit different (more precisely, each matrix $(a_{ij})$ can be "identified" with the vector $(a_{11},a_{12},a_{21},a_{22})$), apply Gaussian elimination to check whether the vectors that span $U$ (and $W$) are linearly independent or not.
Here's an idea about what to do next:
Try to establish $U$ as an equation, i.e., once you have the basis $\{(1,2,4,1),(0,3,1,-1)\}$ of $U$, reduce the matrix
$$\left[\begin{array}{cc|c} 1 & 0 & x \\ 2 & 3 & y\\ 4 & 1 & z \\ 1 & -1 & w\end{array}\right]$$
to find a condition on $(x,y,z,w)$ to be on $U$. Do the same for $W$ and then solve the system of equations you find from there. After that, find a basis for that solution set (Notice we could do this from the beginning without finding a basis for $U$ and $W$, however, this reduces the quantity of computations).