248 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



relative to the given pair of bases. A strong analog of this observation is 

 contained in the following lemma. 



11.13* Lemma: (i) Let L be a geometrical linear correspondence. Fix 

 any real coordinate frame and let [L] be the linear correspondence 

 whose paired n-tuples are 



[M, M, 



where 



[v, k]eLip). 



(ii) Alternatively, let [L] be a (concrete) linear correspondence be- 

 tween w-tuples. Interpret the n-tuples related by [L] as representing 

 vectors in some real coordinate frame. Let L be the geometrical cor- 

 respondence whose pairs, expressed as n-tuples in this frame, are those 

 of the concrete correspondence [L]. 



In either case, (i) or (ii), the geometric correspondence L satisfies the 

 geometric postulates PI , • •• , P7 if and only if the concrete corre- 

 spondence [L] satisfies the concrete forms of these postulates. 



The proof of this lemma consists essentially in reading the postulates 

 carefully. We shall not reproduce it. 



11.2 Our position is now this: We have on the one hand geometrical 

 objects, vectors v, k, operators Z{p), Yip), and geometrical correspond- 

 ences L. On the other hand, we have concrete n-tuples [v], [k], matrices 

 [Z{p)], [Y{p)], and linear correspondences [L]. Given any pair of bases 

 in V and K, in particular, given any coordinate frame, each geometric 

 object generates a corresponding concrete object which represents it 

 relative to those bases or that frame. Conversely, given a concrete ob- 

 ject [^], we can choose a frame in V and K and find that geometric object 

 ^ whose coordinates in the chosen frame are given by [^]. 



11.21* The concrete object, linear correspondence, defines a linear time- 

 invariant 2n-pole by 6.21. To complete the picture, we might say that a 

 geometrical ^^, x-esponden«.e L defines a Cauer class of 2/i-poles. 



11.22* This terminology is motivated by the following observation: 

 if [L] and [L]i are hnear correspondences representing L in two distinct 

 real frames, then there exists a real nonsingular matrix [W] relating the 



M, [kML](p) 



and the 



[Ml , [k]MLUp) 



* Technical paragraph as explained in Section 2.91. 



