A homogeneous system is just a system of linear equations where all constants on the right side of the equals sign are zero.
A homogeneous system always has the solution
This is called the trivial solution. Any nonzero solution is called nontrivial.
has a nontrivial solution
there is a free variable
has a column without a pivot position.
When we row reduce the augmented matrix for a homogeneous system of linear equations, the last column will be zero throughout the row reduction process. We saw this in the last example:
So it is not really necessary to write augmented matrices when solving homogeneous systems.
When the homogeneous equation
does have nontrivial solutions, it turns out that the solution set can be conveniently expressed as a span.
Consider the following matrix in reduced row echelon form:
The matrix equation
corresponds to the system of equations
We can write the parametric form as follows:
We wrote the redundant equations
in order to turn the above system into a vector equation:
This vector equation is called the parametric vector form of the solution set. Since
are allowed to be anything, this says that the solution set is the set of all linear combinations of
In other words, the solution set is
G M K M I E P N
F Q O H M L M J
Here is the general procedure.
matrix. Suppose that the free variables in the homogeneous equation
are, for example,
Put equations for all of the
The solutions to
will then be expressed in the form
for some vectors
and any scalars
This is called the parametric vector form of the solution.
In this case, the solution set can be written as
We emphasize the following fact in particular.
The set of solutions to a homogeneous equation
Since there were two variables in the above example, the solution set is a subset of
Since one of the variables was free, the solution set is a line:
In order to actually find a nontrivial solution to
in the above example, it suffices to substitute any nonzero value for the free variable
For instance, taking
gives the nontrivial solution
Since there were three variables in the above example, the solution set is a subset of
Since two of the variables were free, the solution set is a plane.
There is a natural question to ask here: is it possible to write the solution to a homogeneous matrix equation using fewer vectors than the one given in the above recipe? We will see in example in Section 2.5 that the answer is no: the vectors from the recipe are always linearly independent, which means that there is no way to write the solution with fewer vectors.
Another natural question is: are the solution sets for inhomogeneuous equations also spans? As we will see shortly, they are never spans, but they are closely related to spans.
There is a natural relationship between the number of free variables and the “size” of the solution set, as follows.
The above examples show us the following pattern: when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc. The number of free variables is called the dimension of the solution set.
We will develop a rigorous definition of dimension in Section 2.7, but for now the dimension will simply mean the number of free variables. Compare with this important note in Section 2.5.
Intuitively, the dimension of a solution set is the number of parameters you need to describe a point in the solution set. For a line only one parameter is needed, and for a plane two parameters are needed. This is similar to how the location of a building on Peachtree Street—which is like a line—is determined by one number and how a street corner in Manhattan—which is like a plane—is specified by two numbers.
Recall that a matrix equation
is called inhomogeneous when