On a class of equivalent systems of linear inequalities
Isidore Heller · Pacific Journal of Mathematics · 1963
1.9) d iS = d v = ε^βi + e m+j ) where v = n(i -1) + j e μ denoting the μth unit vector in R m+n .It should be noted that D is of rank m + n -1: first, the subset of columns d il9 d 13 -(i = 1, 2, , m; j = 2, 3, , w) is independent, hence, the rank ^ m + n -1; second, if e -e λ + • +β m -β m+1 -... -e w+% , then consideration of the inner products e f d iό = 0, β' βί = 1 when i ^ m, e'βi = -1 when i > m shows that, say, β x is not representable in terms of columns of D, since e 1 = Σ ^ϋ^ϋ would imply 1 = β ' βl = 2 αiiβ'd^ = 0.For later reference we also mention the geometric interpretation of the set S of columns of D. Setting the e t and f 3 ?(i = 1, 2, , m; j -1, 2, , w), interpreted as points in affine space, are the m + n vertices of an (m + n -1)-simplex and appear thus partitioned into two disjoint sets S={e 4 |i = l,2, --,m}, F -{f d \ j = 1, 2, ... f n}.By (1.9) S consists of all vectors of the form e^fa -/,•), that is of all those edges of the simplex that connect a vertex of E with a vertex of F, the orientation being determined by e iά .In the special case where all ε^ -1 the orientation is always from F to E.