The linear strand of determinantal facet ideals

Let $X$ be an $(m\times n)$-matrix of indeterminates, and let $J$ be the ideal generated by a set $\mathcal{S}$ of maximal minors of $X$. We construct the linear strand of the resolution of $J$. This linear strand is determined by the clique complex of the $m$-clutter corresponding to the set $\mathcal{S}$. As a consequence one obtains explicit formulas for the graded Betti numbers $\beta_{i,i+m}(J)$ for all $i\geq 0$. We also determine all sets $\mathcal{S}$ for which $J$ has a linear resolution.