Page 52 - A Course in Linear Algebra with Applications

36 Chapter Two: Systems of Linear Equations

(vii) Examine equations 3 through m and find the first
one that involves an unknown other than x\ and Xi 2, say x i3.
By interchanging equations we may assumethat Xi 3 actually
occurs in equation 3.

The next step is to make the coefficient of x i3 equal to 1,
and then to eliminate Xi 3 from equations 4 through m, and so
on.
The elimination procedure continues in this manner, pro-
ducing the so-called pivotal unknowns xi = x ix, x i2, ..., Xi r,
until we reach a linear system in which no further unknowns
occur in the equations beyond the rth. A linear system of this
sort is said to be in echelon form; it will have the following
shape.

+ • • • + * x n = *
Xi 1 -f- # Xi 2
+ • • • + * x n = *
Xi 2
< •Ei r i ' ' ' "T * X n — *
0 = *

K 0 = *

Here the asterisks represent certain scalars and the ij are in-
n
tegers which satisfy 1 = i\ < %2 < ••• < r < - The unknowns
i
Xi. for j = 1 to r are the pivots.
Once echelon form has been reached, the behavior of the
linear system can be completely described and the solutions
- if any - obtained by back substitution, as in the preceding
examples. Consequently we have the following fundamental
result which describes the possible behavior of a linear system.

Theorem 2.1.2
(i) A linear system is consistent if and only if all the entries on
the right hand sides of those equations in echelon form which
contain no unknowns are zero.