FORMAL REALIZAHILITY THEORY — I 241 



No\v if reV/,11 , thou 



(r, k) = 



for every /.eK^ = (V/.o)'. Then, liowever, by direel calculalioii from 

 l;5ection 7.0, 



(ir"V, w*k) = 0, 



wlicrc ir* is the adjoint, i.e. tran,si)()se(l conjugate matrix of IT. But 

 because IT is real, IT* = IT'. Hence if reV^i , then 



(W^'v, k) = 



for every Aeir'K^ = K.^ . Hence 



K.„ = (ir-'V;.,,)" = (YUp))'. 



By this and its dual, P3 is completed for M . 



The reamining postulates for M follow from those for L by the simple 

 equality 



(v, k) = (W'\, W'k) 



already established, combined with r,,/ = r^ . 



9.32 For fi.xed Z(p), the matrices (1), as U ranges over a group, form an 

 ecjui valence class. Classical matrix theory treats of such equivalence 

 classes. This author's predilection is to regard this theory from a geo- 

 metrical point of view. In part this prejudice may be justified by the 

 ease with which that slightly more general object, a linear corre- 

 spondence, can be treated by geometrical methods. In any event we shall 

 begin our program of proofs with a lirief introduction to the geometrical 

 approach. 



X. GEOMETRICAL PRELIMINARIES 



10.0* We now wish to consider V and K as complex n-dimensional 

 linear spacesj respectively of voltage vectors v and current vectors k. 

 The distinction here is in point of view. A vector v is regarded as an 

 absolute geometrical object; an n-tuple [v] = [ai , • • • , a„] is regarded 

 as a set of coordinates for the vector v, relative to some coordinate basis. 

 Given a fixed coordinate basis, there is a one-to-one correspondence 

 between vectors v and the n-tuples [v] which represent them in that 

 basis, a correspondence which preserves the operations of vector algebra. 



* Technical paragraph as e.\plained in Section 2.91. 



t For a reference concerning the ideas in this section, see Halmos^ Chapters 

 I and II. 



