  Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring
over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free
resolution of $J$ and show that it is supported on a polymatroid obtained from
the underlying representable polymatroid by means of the so-called Dilworth
truncation. Formulas for the projective dimension and Betti numbers are given
in terms of the polymatroid as well as a characterization of the associated
primes. Along the way we show that $J$ has linear quotients. In fact, we do
this for a large class of ideals $J_P$, where $P$ is a certain poset ideal
associated to the underlying subspace arrangement.
