Page 54 - A Course in Linear Algebra with Applications

38 Chapter Two: Systems of Linear Equations

Here the pivots are x\ and X3. One further operation must
be applied to put the system in reduced row echelon form,
namely (1) - 3(2); this gives

x\ + 3x2 + X4 — — 2
£3 + \x± = 1

To obtain the general solution give the non-pivotal unknowns
and X4 the arbitrary values d and c respectively, and then
x 2
read off directly the values xi = — 2 — c — 3d and x 3 = 1 — c/3.

Homogeneous linear systems
A very important type of linear system occurs when all
the scalars on the right hand sides of the equations equal zero.

' a n x i + CJ12X2 +
a 2\Xi + a 22x 2 + + «2n^n = 0

+ + =
, a mlx 1 a m2x 2 1 Q>mn%n U

Such a system is called homogeneous. It will always have the
trivial solution x\ = 0, x 2 = 0, ..., x n = 0; thus a homogeneous
linear system is always consistent. The interesting question
about a homogeneous linear system is whether it has any non-
trivial solutions. The answer is easily read off from the echelon
form.

Theorem 2.1.3
A homogeneous linear system has a non-trivial solution if and
only if the number of pivots in echelon form is less than the
number of unknowns.
For if the number of unkowns is n and the number of
pivots is r, the n — r non-pivotal unknowns can be given arbi-
trary values, so there will be a non-trivial solution whenever