Page 12 - Matrix Analysis & Applied Linear Algebra

4 Chapter 1 Linear Equations

and write the system as
 
 E 1 
 
E 2
 
S = . .
 . 
 . 
 
E m
For a linear system S , each of the following three elementary operations
results in an equivalent system S .

(1) Interchange the i th and j th equations. That is, if

   
E 1 E 1
   
   
 .   . 
 .   . 
   
 .   . 
   
   
   
 E i   E j 
. .
   
S = . . , then S = . . . (1.2.1)

   
   
  
 E j 
  E i 
   
   
 .   . 
 .   . 
   
 .   . 
   
E m E m
(2) Replace the i th equation by a nonzero multiple of itself. That is,
 
 E 1 
 
 . 
 . 
 
 . 
S = αE i , where α =0. (1.2.2)

 
 . 
 . 
 
 . 
 
E m
(3) Replace the j th equation by a combination of itself plus a multiple of
the i th equation. That is,
 
E 1
 
 . 
 . 
 
 . 
 
 
 
 
 E i 
.
 
S = . . . (1.2.3)

 
 


 E j + αE i 
 
 
 . 
 . 
 . 
 
 
E m