278 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1952 



where the first r span Vi , the last n — r, Yo. Let 



Vs = Ifs + iWs , 1 < S < 71, 



where u, , uh are real (10.42). Since Vi is real and a linear manifold, 



^is = K^s + Vs)eYi , 1 < s < /•, 

 and, similarlj^, WseVi , I < s < r. Among the 2/i real vectors 



the first 2r are in Vi , and they span Vi because the i'^ , 1 < s < r, can 

 be constructed from them. The whole list (1) spans V, because from it 

 all the Vs , I < s < n, can be constinicted. Since the VseV-i do not use 

 in then- construction any of the first 2r vectors (1), it follows that the 

 last 2(n — r) vectors in that hst must contain a set spanning V2 . The 

 realit}' of the vectors (1) then establishes the existence of a real basis, 

 say, 



v[ , ■ • • ,Vr , Vr+l , • • • ,Vn (2) 



which provides a basis in Vi and V2 . 



20.11 We now have 7.22. The unique dual basis 



h' ... h' 



to (2) is real by 10.41. Hence all of Vi , V2 , Ki , K2 are real. The proof 

 of 10.6 is then complete. 



20.2 If in a real basis (2) (dropping primes) 



V = ail'i + a2l'2 + • • • + dnVn , 



that is, if 



[v] = [ai , • • • , Gn], 

 then by (5) of 10.3 



V = diVi + • • • + dnVn , 



hence 



[v] = [ai , • • • , Qn]. 



The geometrical conjugation of 10.3 is therefore simply the concrete 

 one of 7.2 in any real basis. This proves the remark of 10.35. 



