Definitions: Flock, Herd Space, Linear Flock

Proposition 8.1.6: The herd space of a flock F contains three functions based at non-collinear points that are scalar multiples of each other, if and only if, F is a linear flock.

Pf: The necessity of the condition follows immediately from the characterization of the herd space of a linear flock given in Proposition 8.1.4 [backbutton].

To prove the sufficiency, we may assume that fQ = ufP, and fR = vfP for three non-collinear points P, Q and R. If fP is the constant function, then F is linear by Proposition 8.1.4 [backbutton], so we assume that it is not constant. Similarly, we may assume that u and v are not both zero. Now, since three scalar multiples of the same function are linearly dependent, we have by Proposition 8.1.5 [backbutton], that F is a star flock. To prove that F is linear, we need to show that there are at least two sets of constants, a,b,c and a',b',c' with the second set not a scalar multiple of the first set, which express the linear dependence of the functions, i.e. afP + bfQ + cfR = 0 and a'fP + b'fQ + c'fR = 0. This follows from the proof of Proposition 8.1.5 [backbutton], since each such set gives rise to a common point on all the planes of the flock, and only sets which are scalar multiples of each other give rise to the same point. Since we have,

vfP + 0fQ - fR = 0, and
ufP - fQ + 0fR = 0,
and no scalar multiple of (u, -1, 0) can equal (v, 0, -1), there are at least two common points on all planes of F, and so F is linear.

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