Linear Algebra

Section I. Solving Linear Systems 27
Proof. We’ve seen examples of all three happening so we need only prove that
those are the only possibilities.
First, notice a homogeneous system with at least one non-�0 solution �v has
infinitely many solutions because the set of multiples s�v is infinite — if s �= 1
then s�v − �v =(s − 1)�v is easily seen to be non-�0, and so s�v �= �v.
Now, apply Lemma 3.8 to conclude that a solution set
{�p + � h � � � h solves the associated homogeneous system}
is either empty (if there is no particular solution �p), or has one element (if there
is a �p and the homogeneous system has the unique solution �0), or is infinite (if
there is a �p and the homogeneous system has a non-�0 solution, and thus by the
prior paragraph has infinitely many solutions). QED
This table summarizes the factors affecting the size of a general solution.
particular
solution
exists?
yes
no
number of solutions of the
associated homogeneous system
one infinitely many
unique
solution
infinitely many
solutions
no
solutions
no
solutions
The factor on the top of the table is the simpler one. When we perform
Gauss’ method on a linear system, ignoring the constants on the right side and
so paying attention only to the coefficients on the left-hand side, we either end
with every variable leading some row or else we find that some variable does not
lead a row, that is, that some variable is free. (Of course, “ignoring the constants
on the right” is formalized by considering the associated homogeneous system.
We are simply putting aside for the moment the possibility of a contradictory
equation.)
A nice insight into the factor on the top of this table at work comes from considering
the case of a system having the same number of equations as variables.
This system will have a solution, and the solution will be unique, if and only if it
reduces to an echelon form system where every variable leads its row, which will
happen if and only if the associated homogeneous system has a unique solution.
Thus, the question of uniqueness of solution is especially interesting when the
system has the same number of equations as variables.
3.12 Definition A square matrix is nonsingular if it is the matrix of coefficients
of a homogeneous system with a unique solution. It is singular otherwise,
that is, if it is the matrix of coefficients of a homogeneous system with infinitely
many solutions.