muli 
and 
n-tu 
§ 4j 
(23) 
as 
£i«i 
form 
assui 
of n 
stani 
The 
ditio; 
Undi 
a sys 
and 
lates 
Ur 
order 
of F, 
and 
impos 
deterr 
a defi 
units 
algebr 
is a fin: 
22 INTRODUCTION, DEFINITIONS [chap, i 
From these relations we obtain again (15), aside from 
the lettering of the subscripts, if we write e rs for e' sr , i.e., if 
we interchange the rows and columns of our matrices. 
Hence the algebra of all /»-rowed matrices over F is self 
reciprocal under the correspondence which interchanges 
the rows and columns of its matrices. 
Two algebras which are either both equivalent or 
both reciprocal to the same algebra are equivalent to 
each other. 
13. Second definition of an algebra. Each element 
x = S£iUi of an algebra A over F, defined in § 4, has a 
unique set of co-ordinates . . . . , in F with 
respect to a chosen set of basal units u 1} . . . . , u n . 
Hence with x may be associated a unique w-tuple* 
[£i, ....,£«] of n ordered numbers of F. Using 
this n-tuple as a symbol for x, we may write equations 
(iOj), (13), (14) in the following form: 
(20) £»]+hi, 
(21) [£ r , ....,£»]• hi, ,Vn] 
~ 71 n 
= ^ ^ £iVj'iiji )••••) ^iVj'Yijn ) 
_i,j — I = i 
(22) p[£j 5 . £»l = [£i, , £«]p 
= [p£i, • • • • 5 p£«L p ^ T. 
These preliminaries suggest the following definition 
by W. R. Hamilton of an algebra A over F: Choose 
any n 3 constants 7of F, consider all n-tuples [£ l5 . . . . , 
£J of n ordered numbers of F, and define addition and 
* For an algebra of two-rowed matrices, the numbers of each quad 
ruple were written by twos in two rows.