  We construct the first linear strand of the minimal free resolutions of edge
ideals of $d$-partite $d$-uniform clutters. We show that the first linear
strand is supported on a relative simplicial complex. In the case that the edge
ideals of such clutters have linear resolutions, we give an explicit and
surprisingly simple description of their minimal free resolutions, generalizing
the known resolutions for edge ideals of Ferrers graphs and hypergraphs and
co-letterplace ideals. As an application, we show that the Lyubeznik numbers
that appear on the last column of the Lyubeznik table of the cover ideal of
such clutters are Betti numbers of certain simplicial complexes. Furthermore,
we restate a characterization for edge ideals of $d$-partite $d$-uniform
clutters which have linear resolutions based on the recent characterization of
arithmetically Cohen-Macaulay sets of points in a multiprojective space.
