Let $v=(1,1,1,0,0)$ and $w=(1,0,0,-1,-1)$ be vectors in $\mathbb{R}^5$. Find a basis and the dimension for the subspace $S=\{u \in \mathbb{R}^5 \mid u \cdot v=0 \text{ and } u\cdot w = 0\}$.
I tried putting them into a matrix and reducing, but I have no idea where to go from there. It is an assignment question, so I'm not asking for the answer, just and idea of how to solve it, because I have nothing.
Hint: If you write out the equations that define $S$, you'll see that $u = (u_1,\dots,u_5)$ is an element of $S$ exactly when it is a solution to the system $$ \pmatrix{ 1&1&1&0&0\\ 1&0&0&-1&-1 } \mathbf{u} = \pmatrix{0\\0} $$