*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 = (x_{P},y_{P},z_{P},0), Q = (x_{Q},y_{Q},z_{Q},0) and R = (x_{R},y_{R},z_{R},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 f_{P}(t) = f(t)x_{P} + g(t)y_{P} + h(t)z_{P}, f_{Q}(t) = f(t)x_{Q} + g(t)y_{Q} + h(t)z_{Q} and f_{R}(t) = f(t)x_{R} + g(t)y_{R} + h(t)z_{R}. For scalars a, b and c we have:

af_{P}(t) + bf_{Q}(t) + cf_{R}(t) = f(t)[ax_{P} + bx_{Q} + cx_{R}] + g(t)[ay_{P} + by_{Q} + cy_{R}] + h(t)[az_{P} + bz_{Q} + cz_{R}].
Noticing that (a,b,c,0)M =(ax_{P} + bx_{Q} + cx_{R}, ay_{P} + by_{Q} + cy_{R}, az_{P} + bz_{Q} + cz_{R}, 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, (ax_{P} + bx_{Q} + cx_{R}, ay_{P} + by_{Q} + cy_{R}, az_{P} + bz_{Q} + cz_{R}, 0) represents a point in W = 0. Thus, we see that the functions f_{P}, f_{Q} and f_{R} 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 af_{P} + bf_{Q} + cf_{R} = 0 for all t in GF(q)) iff there exists a point ((ax_{P} + bx_{Q} + cx_{R}, ay_{P} + by_{Q} + cy_{R}, az_{P} + bz_{Q} + cz_{R}, 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.

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