234 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



Then its annihilator (Vi)" is the set of all k such that 



t'eVi implies (v, k) — 0. 



(Vi) is a linear manifold in K. 



Dually, given Ki C K, (Ki)" is the linear manifold of all reV such 

 that 



/ceKi implies (v, k) = 0. 



The annihilator concept is the analog in our general geometric frame- 

 work of the idea of orthogonalit,y. It clearly suggests a connection with 

 workless constraints. 



7.2* The complex conjugate of an /i-tuple v (or /.•) is defined in the 

 obvious way: if 



V = [vi , • ■ ■ , Vn] 

 then 



V = [Vi , ■ • ■ , Vn]. 



This conjugation operation clearlj^ has the properties 



I = ^ 



(2) 



a^ -{- br] = a^ -\- hfj 



where a and h are scalars and ^ and rj are (consistently) elements of V 

 or K. Furthermore, at once from (1) of 7.0, 



(v, k) = (v, k). (3) 



7.21* A linear manifold will be called real if it contains, with any 

 n-tuple also the conjugate of that n- tuple. 



7.22* A real manifold is spanned by real 7i-tuples. This will be proved 

 in the Appendix, Section 20. 



7.23* The annihilator of a real manifold is real. For let Ki be real and 

 k , • • • , k^ he real n-tuples which span Ki . Then if ve (Ki) every 



(v, k') = 0, 



and conversely. But then also 



(v, k') = {v, k') = = 0, 

 so i;6(Ki)''. 



* Technical paragraph as explained in Section 2.91. 



