If you have two linear subspaces $V_1$ and $V_2$ of a vector space $V,$ both given by their bases, there is fairly heavy handed way of computing their intersection: write down the projection matrices onto both subspaces, then multiply them, then compute the column span by row reduction. This is $O(n^3),$ where $n$ is the dimension of $V,$ even if $V_1$ and $V_2$ are (e.g.) two dimensional, but has the advantage of being very easy to code.

Is there a faster elegant method?