8. Basis for an intersection of subspaces

Problem. Find a basis for the intersection in $\mathbb{R}^3$ of the subspace $W_1$ spanned by $(1,2,1)$, $(2,4,2)$, and $(1,3,3)$ and the subspace $W_2$ spanned by $(2,7,8)$ and $(3,8,8)$.

Method. We can't use basis elements from $W_1$ and $W_2$, since they are almost certainly outside the intersection. Instead, we can use an indirect method: Find homogeneous linear equations whose solution space is $W_1$ (or in other words, find the rows of a matrix whose null space is $W_1$, as in Section ), and similarly for $W_2$. Taking both sets of equations together, the solution space will be $W_1 \cap W_2$. Then find a basis for this solution space.