Page 55 - A Course in Linear Algebra with Applications

2.1: Gaussian Elimination 39

n — r > 0. On the other hand, if n = r, none of the unknowns
can be given arbitrary values, and there is only one solution,
namely the trivial one, as we see from reduced echelon form.

Corollary 2.1.4
A homogeneous linear system of m equations in n unknowns
always has a non-trivial solution if m <n.

For if r is the number of pivots, then r <m < n.
Example 2.1.5

For which values of the parameter t does the following homo-
geneous linear system have non-trivial solutions?

6a?i - x 2 + x 3 = 0
tX\ + X3 = 0
+ = 0
x 2 tx 3
It suffices to find the number of pivotal unknowns. We
proceed to put the linear system in echelon form by applying
|
to it successively the operations (1), (2) — £(1), (2) «->• (3)
and ( 3 ) - | ( 2 ) :

{ Xl - \x 2 + | ^ 3 = 0

+ = 0
x 2 tx 3
U - S - T ) * 3 =0
The number of pivots will be less than 3, the number of un-
2
t
knowns, precisely when 1 — t/6 — /6 equals zero, that is,
when i = 2 or i = - 3 . These are the only values of t for
which the linear system has non-trivial solutions.
The reader will have noticed that we deviated slightly
from the procedure of Gaussian elimination; this was to avoid
dividing by t/6, which would have necessitated a separate dis-
cussion of the case t = 0.