Intersection of vector subspaces

Dear All,

I hope you can help me solving the following problem.

Given several subspaces $\displaystyle S_i$ of a linear $\displaystyle n$-dimensional vector space, each subspace being determined by a set of $\displaystyle n_i$ $\displaystyle (n_i\le n)$ linearly independent vectors, find a set of vectors spanning the intersection of the subspaces.

It suffices to solve the problem just for two subspaces, as the rest can be done iteratively. Is there any simple method for determining the dimension of the intersection in question without finding the vectors?

Thank you for any hint.