Page 53 - A Course in Linear Algebra with Applications

2.1: Gaussian Elimination 37

(ii) If the system is consistent, the non-pivotal unknowns can
be given arbitrary values; the general solution is then obtained
by using back substitution to solve for the pivotal unknowns.
(iii) The system has a unique solution if and only if all the
unknowns are pivotal.

An important feature of Gaussian elimination is that it
constitutes a practical algorithm for solving linear systems
which can easily be implemented in one of the standard pro-
gramming languages.

Gauss-Jordan elimination
Let us return to the echelon form of the linear system
described above. We can further simplify the system by sub-
tracting a multiple of equation 2 from equation 1 to eliminate
from that equation. Now occurs only in the second
x i2 x i2
equation. Similarly we can eliminate x; 3 from equations 1
and 2 by subtracting multiples of equation 3 from these equa-
tions. And so on. Ultimately a linear system is reached which
is in reduced echelon form.
Here each pivotal unknown appears in precisely one equa-
tion; the non-pivotal unknowns may be given arbitrary values
and the pivotal unknowns are then determined directly from
the equations without back substitution.
The procedure for reaching reduced echelon form is called
Gauss-Jordan elimination: while it results in a simpler type of
linear system, this is accomplished at the cost of using more
operations.

Example 2.1.4
In Example 2.1.3 above we obtained a linear system in echelon
form

' X\ + 3^2 + 3^3 + 2^4 = 1
|
< x 3 + x 4 = 1