[chap. I 
§ 14] COMPARISON OF THE TWO DEFINITIONS 23 
aside from 
r 6 S ri EC., if 
r matrices. 
;r F is self- 
iterchanges 
multiplication of n-tuples by formulas (20) and (21), 
and scalar multiplication of a number of p of F and an 
w-tuple by formula (22). To pass to the definition in 
§ 4, employ the particular w-tuples 
(23) «i=[l, 0, . . . . , 0], u 2 = {0, 1, 0, . . . . , 0], . . . . , 
livalent or 
uivalent to 
«« = [0, . . . . , 0, 1] 
as basal units. By (20) «and (22), [&, = 
ich element 
i § 4, has a 
in F with 
. . . . , u n , 
ae w-tuple* 
F. Using 
te equations 
£iMi+ .... -\-% n u n . Then (20), (21), (22) take the 
form ( 1 Oj), (13), (14), and, as noted in § 9, all of the 
assumptions made in § 4 are satisfied. Hence an algebra 
of w-tuples is an algebra according to § 4 and conversely. 
Hence there exists an algebra over F having as con 
stants of multiplication any given n 3 numbers 7of F. 
The algebra will be associative if the 7’s satisfy the con 
ditions (§58) obtained from {uiUj)u k = Ui{ujU k ). 
14. Comparison of the two definitions of an algebra. 
Under the definition in § 4, an algebra over a field F is 
. • , > 
a system consisting of a set of wholly undefined elements 
and three undefined operations which satisfy five postu 
n 
kiVj'Yijn ) 
i,j = i 
lates. 
Under Hamilton’s definition in § 13, an algebra of 
order n over F is a system consisting of n 3 constants 7 t j k 
of F, a set of partially* defined elements [£ r , . . . . , £ w ], 
ini P in F - 
and three defined operations, while no postulates are 
imposed on the system other than that which partially 
ng definition 
r F: Choose 
es [&, . . . . , 
addition and 
determines the elements. This definition really implies 
a definite set (23) of basal units. A transformation of 
units leads to a new algebra (equivalent to the initial 
algebra) with new values for the n 3 constants 7^. 
;rs of each quad- 
* Each element is an w-tuple of numbers of F. In particular, if F 
is a finite field of order p, there are evidently exactly p n elements.