Definitions: Flock, Herd Space, Star Flock

Proposition 8.1.5: The herd space of a flock F contains three functions based at non-collinear points that are linearly dependent over GF(q), if and only if, F is a star flock and all triples of functions based at non-collinear points are linearly dependent over GF(q).

Pf: Let P = (xP,yP,zP,0), Q = (xQ,yQ,zQ,0) and R = (xR,yR,zR,0) be three non-collinear points of W = 0. The non-collinearity of the three points is equivalent to the statement that the matrix

is non-singular. If F = F(f,g,h) is a flock, then the herd space functions at these three points are fP(t) = f(t)xP + g(t)yP + h(t)zP, fQ(t) = f(t)xQ + g(t)yQ + h(t)zQ and fR(t) = f(t)xR + g(t)yR + h(t)zR. For scalars a, b and c we have:

afP(t) + bfQ(t) + cfR(t) = f(t)[axP + bxQ + cxR] + g(t)[ayP + byQ + cyR] + h(t)[azP + bzQ + czR].
Noticing that (a,b,c,0)M =(axP + bxQ + cxR, ayP + byQ + cyR, azP + bzQ + czR, 0), since M is non-singular, the right hand side is the zero vector iff the left hand side is the zero vector (i.e., a, b and c are all zero). So, if a, b and c are not all zero, (axP + bxQ + cxR, ayP + byQ + cyR, azP + bzQ + czR, 0) represents a point in W = 0. Thus, we see that the functions fP, fQ and fR are linearly dependent for any choice of three non-collinear points (i.e., there exist constants a, b and c, not all zero, so that afP + bfQ + cfR = 0 for all t in GF(q)) iff there exists a point ((axP + bxQ + cxR, ayP + byQ + cyR, azP + bzQ + czR, 0)) in each of the planes with equations, f(t)X + g(t)Y + h(t)Z + W = 0. That is to say, iff F is a star flock.