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2.4 Homogeneous Systems 57
2.4 HOMOGENEOUS SYSTEMS
A system of m linear equations in n unknowns
a 11 x 1 + a 12 x 2 + ··· + a 1n x n =0,
a 21 x 1 + a 22 x 2 + ··· + a 2n x n =0,
.
.
.
a m1 x 1 + a m2 x 2 + ··· + a mn x n =0,
in which the right-hand side consists entirely of 0’s is said to be a homogeneous
system. If there is at least one nonzero number on the right-hand side, then the
system is called nonhomogeneous. The purpose of this section is to examine
some of the elementary aspects concerning homogeneous systems.
Consistency is never an issue when dealing with homogeneous systems be-
cause the zero solution x 1 = x 2 = ··· = x n = 0 is always one solution regardless
of the values of the coefficients. Hereafter, the solution consisting of all zeros is
referred to as the trivial solution. The only question is, “Are there solutions
other than the trivial solution, and if so, how can we best describe them?” As
before, Gaussian elimination provides the answer.
While reducing the augmented matrix [A|0] of a homogeneous system to
a row echelon form using Gaussian elimination, the zero column on the right-
hand side can never be altered by any of the three elementary row operations.
That is, any row echelon form derived from [A|0] by means of row operations
must also have the form [E|0]. This means that the last column of 0’s is just
excess baggage that is not necessary to carry along at each step. Just reduce the
coefficient matrix A to a row echelon form E, and remember that the right-
hand side is entirely zero when you execute back substitution. The process is
best understood by considering a typical example.
In order to examine the solutions of the homogeneous system
x 1 +2x 2 +2x 3 +3x 4 =0,
2x 1 +4x 2 + x 3 +3x 4 =0, (2.4.1)
3x 1 +6x 2 + x 3 +4x 4 =0,
reduce the coefficient matrix to a row echelon form.
1223 1 2 2 3 1 2 2 3
A = 2413 −→ 00 −3 −3 −→ 0 0 −3 −3 = E.
3614 00 −5 −5 0 0 0 0
Therefore, the original homogeneous system is equivalent to the following reduced
homogeneous system:
x 1 +2x 2 +2x 3 +3x 4 =0,
(2.4.2)
− 3x 3 − 3x 4 =0.
