FORMAL REALIZABILITY THEORY — I 255 



])(' the basis vectors spanning; Ko . By 12.32, there are iinif|uc vectors 

 "lip), ■ ■ ■ , Umiv) i» V2 such that 



Let?^i , • • • , Vm be the (real) basis xcclors in Vjdual to the Ai , ■ • • , /,„, : 

 (iV , A\) = drs 1 < '• < s. (24) 



Since the i(r(p) are all in V.. we hav(> for each pel\ 



Ws(p) = S ars{p)vr (25) 



r = l 



wIkm'c the coefficients Orsip) are calculated by (24) to l)e 



orsip) = Mp), a-,.). (2G) 



12.41 Because the Av are real, P5(I) implies that 



Usrip) = (Urip), ks) = Mp), kr) = ttrsip) . (27) 



By the reasoning just following (8) and by the uniqueness of the 

 Ws(p)€V2 , since V2 is real, we have Us(p) = Ws(p). Then 



arsip) = (Usip), kr) = (Usip), kr) = ttrsip). 



12.42 We have by P2 that 



[urip) + \K.ip), kr + Xks]eLip), (28) 



for any X. The identity 



illr + Vs , kr + ks) - (Wr " 1h , K - A',) 



= 2(w, , A-.) + 2(W3 , kr) 



(29) 



holds in fact for any vectors Ur , u^ , Av , A-^ . Using (27), (28) and P6(I), 

 it exhibits Orsip) as a rational function. 



12.5 Consider the »i X m matrix [Ziip)] whose elements are the arsip). 

 the s-th column of this matrix consists of the components of «.,(p). 

 The rank of the matrix is by definition the dimension of the space 

 spanned by these columns. 



12.51 Now the rank of [Ziip)] can be expressed in terms of the vanish- 

 ing or not of its various minor determinants. There are finitely many 

 such minors and each is a rational function. Each is either identically 

 zero or else Nanishes at only finitely many points. Hence the rank of 

 [Ziip)], except at these finitely many points, and at the p in the comple- 



