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.

Since there were four 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.

Recipe: Parametric vector form (homogeneous case)

Let be an matrix. Suppose that the free variables in the homogeneous equation are Then the solutions to can be written in the form

for some vectors in and any scalars This is called the parametric vector form of the solution. It is obtained by finding the parametric form of the solution, including the redundant equations putting these equations in order, and making a vector equation.

In this case, the solution set can be written

Note that the solution set of a homogeneous equation is a span!

Subsection3.4.2Inhomogeneous Systems

Recall that a matrix equation is called inhomogeneous when

In the above example, the solution set was all vectors of the form

where and are any scalars. In this case, a particular solution is

In the previous example and the example before it, the parametric vector form of the solution set of was exactly the same as the parametric vector form of the solution set of (from this example and this example, respectively), plus a particular solution.

Key Observation

The set of solutions to if it is nonempty, is obtained by taking one particular solution of and adding all solutions of

In particular, the solution set of is either empty, or it is a translate of a span.

The parametric vector form of the solutions of is just the parametric vector form of the solutions of plus a particular solution

It is not hard to see why the key observation is true. If is a particular solution, then and if is a solution to the homogeneous equation then

As in the first subsection, when there is one free variable in a consistent matrix equation, the solution set is a line—this line does not pass through the origin when the system is inhomogeneous—when there are two free variables, the solution set is a plane, etc.

We will develop a rigorous definition of dimension in Section 3.7, but for now it is important to note that the “dimension” of the solution set of a consistent system is equal to the number of free variables.

Subsection3.4.3Solution Sets and Column Spans¶ permalink

To every matrix we have now associated two completely different geometric objects, both described using spans.

The solution set: for fixed this is the set of all such that

This is a span if and it is a translate of a span (or it is empty) if

It is a subset of

It is computed by solving a system of equations: usually by row reducing and finding the parametric vector form.

The span of the columns of : this is the set of all such that is consistent.

This is always a span.

It is a subset of

It is not computed by solving a system of equations: row reduction plays no role.