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7.5 Homogeneous Systems 213

⎛ ⎞
−4 −1 1 6 −3 2 1 1 0
12. ⎝ 0 4 −4 2 ⎠ 14.
6 −4 −2 −2 0
1 0 0 0
⎛ ⎞
−2 5 7 15. Let A be any matrix of real numbers. Prove that
13. ⎝ 0 1 −3 ⎠
t
rank(A) = rank(A ).
−4 11 11

7.5 Homogeneous Systems

We want to develop a method for finding all solutions of a linear homogeneous system of
n equations in m unknowns:
a 11 x 1 + a 12 x 2 + ··· + a 1m x m = 0
a 21 x 1 + a 22 x 2 + ··· + a 2m x m = 0
.
.
.
a n1 x 1 + a n2 x 2 + ··· + a nm x m = 0.
The numbers a ij are called the the coefficients of the system and A =[a ij ] is the matrix
of coefficients.Row i contains the coefficients of equation i and column j contains the
coefficients of x j .

Define
⎛ ⎞
x 1
x 2
⎜ ⎟
⎜ ⎟
X = ⎜ . ⎟
.
⎝ . ⎠
x m
and write the n × 1 zero matrix as just O, a column of n zeros. Then the system can be written
as the matrix equation
AX = O.
We will develop the following strategy for solving this system.
1. We will show that AX = O has the same solutions as the reduced system A R X = O.
2. We will show how to write all solutions of the reduced system directly from the reduced
matrix A R .
3. We will also use facts about vector spaces and rank to derive additional information about
solutions.
The remainder of this section consists of the details of carrying out this strategy, and
examples. The first two examples give us some feeling for what to look for in solving a
homogeneous system.

EXAMPLE 7.18
Consider the simple system
x 1 − 3x 2 + 2x 3 = 0

−2x 1 + x 2 − 3x 3 = 0.

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October 14, 2010 14:23 THM/NEIL Page-213 27410_07_ch07_p187-246