PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 



51 



Matrices in Hyperspace. 



5. We shall call the order of a space the number of homogeneous 

 coordinates of a point in that space.* Thus a point is of order one, 

 a line of order two, etc. A space of order n can be generated either 

 by points or by hyperplanes of order n-1. A space R of order r < n 

 can be determined by a set of r points A, giving rise to a matrix. 



R = 



an ai2- • • -Oin 



O21 W22 • ■ • •Cf2n 



= [A,A2..M- 



This matrix represents the space R in the sense that from the matrix 

 can be determined the coordinates of the space. The same space R 

 can be determined as the intersection of n-r hyperplanes ai determin- 

 ing a matrix 



O-r+l, 1 Clr+i, 2 • • • -Ctr+l, n 



C<T+2, 1 ^r+2, 2 • ■ ■ • ^r+2, n 



0-n 1 



0-n, 2- 



'■n, n 



[O-r+l O.r+2 



.aj. 



The condition that these matrices represent the same space is that in 

 the determinant 



«11 «12 ttln 



0-n,l 



the minors in the first r rows be proportional to their algebraic compli- 

 ments.^ If in the determinant each minor of the first r rows is 

 equal to its algebraic compliment we shall write 



Ui A. 



A,] — [a^+i ar+2- ■ ■ -ctj- 



The r-rowed determinants of a matrix of r rows we call the elements 

 of the matrix. To add such matrices we add corresponding elements. 

 If there exists a matrix of n columns whose elements are the corre- 



4 Cf. Whitehead, Universal Algebra, page 177. 



5 Cf. Bertini, Geometria Projettiva, p. 33. 



