Since each element of R x or S x is linear and homo 
geneous in the co-ordinates & of x by (6) or (7), we have 
Rax &Rx 
R x ~\~ Ry Rx-\-y 1 
for every number a of F, and the similar equations in S. 
By (8j) and (9), the correspondence between elements 
x, y, .... of algebra A and matrices R x , R y , .... is 
such that xy, ax, and x+y correspond to RxR yi aR x , and 
R x +R y , respectively. Moreover, if A has a modulus, 
this correspondence is one-to-one. For, if R x = R y , then 
o = R x —R y = R x -y, whence x—y = o by Theorem 1. 
Hence by § 12 we have 
Theorem 2. Any associative algebra A with a modulus 
is equivalent to the algebra whose elements are the first 
matrices R x of the elements x of A, and is reciprocal to 
the algebra whose elements are the second matrices S x of 
the elements x of A. 
For example, let A be the algebra of two-rowed 
matrices 
m = 
M = 
'<*! ft 
* bj- 
Then [i^mii and ¡x 1 = ixm lead to transformations 
T m on the variables a, y, (3, d, and t m on a, /5,7, 5, having 
the matrices 
Rm = 
S„ 
where R m is with respect to the units e 11} e 2X , e I2 , e 22 of 
§ 8, and S m is with respect to e ll7 e 12 , e 2I , e 22 . By inspec- 
96 CHARACTERISTIC, RANK EQUATIONS [chap, vii