We have a look at the homogeneous version of
example,
so we consider the homogeneous linear system
-
over
. Due to
fact,
the solution set
is a linear subspace of
. We have described it explicitly in
example
as
-
which also shows that the solution set is a vector space. With this description, it is clear that
is in bijection with
, and this bijection respects the addition and also the scalar multiplication
(the solution set
of the inhomogeneous system is also in bijection with
, but there is no reasonable addition nor scalar multiplication on
).
However, this bijection depends heavily on the chosen "basic solutions“
and
,
which depends on the order of elimination. There are several equally good basic solutions for
.