Linear Algebra

Section I. Solving Linear Systems 21
combination of no vectors). A zero-element solution set fits the pattern since
there is no particular solution, and so the set of sums of that form is empty.
We will show that the examples from the prior subsection are representative,
in that the description pattern discussed above holds for every solution set.
3.1 Theorem For any linear system there are vectors � β1, ... , � βk such that
the solution set can be described as
{�p + c1 � β1 + ··· + ck � �
βk � c1, ... ,ck ∈ R}
where �p is any particular solution, and where the system has k free variables.
This description has two parts, the particular solution �p and also the unrestricted
linear combination of the � β’s. We shall prove the theorem in two
corresponding parts, with two lemmas.
We will focus first on the unrestricted combination part. To do that, we
consider systems that have the vector of zeroes as one of the particular solutions,
so that �p + c1 � β1 + ···+ ck � βk can be shortened to c1 � β1 + ···+ ck � βk.
3.2 Definition A linear equation is homogeneous if it has a constant of zero,
that is, if it can be put in the form a1x1 + a2x2 + ··· + anxn =0.
(These are ‘homogeneous’ because all of the terms involve the same power of
their variable — the first power — including a ‘0x0’ that we can imagine is on
the right side.)
3.3 Example With any linear system like
3x +4y =3
2x − y =1
we associate a system of homogeneous equations by setting the right side to
zeros.
3x +4y =0
2x − y =0
Our interest in the homogeneous system associated with a linear system can be
understood by comparing the reduction of the system
3x +4y =3
2x − y =1
−(2/3)ρ1+ρ2
−→
3x + 4y =3
−(11/3)y = −1
with the reduction of the associated homogeneous system.
3x +4y =0
2x − y =0
−(2/3)ρ1+ρ2
−→
3x + 4y =0
−(11/3)y =0
Obviously the two reductions go in the same way. We can study how linear systems
are reduced by instead studying how the associated homogeneous systems
are reduced.