In this section we will study the geometry of the solution set of any matrix equation
The equation is easier to solve when so we start with this case.
A system of linear equations of the form is called homogeneous.
A system of linear equations of the form for is called inhomogeneous.
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.
The equation 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 and 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 and are allowed to be anything, this says that the solution set is the set of all linear combinations of and In other words, the solution set is
Here is the general procedure.
Let be an matrix. Suppose that the free variables in the homogeneous equation are, for example, and
The solutions to will then be expressed in the form
for some vectors in 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 is a span.
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 Compare to this important note in Section 2.3.
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 3.2 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 3.4, but for now the dimension will simply mean the number of free variables. Compare with this important note in Section 3.2.
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
In the above example, the solution set was all vectors of the form
where is any scalar. The vector is also a solution of take We call a particular solution.
In the solution set, is allowed to be anything, and so the solution set is obtained as follows: we take all scalar multiples of and then add the particular solution to each of these scalar multiples. Geometrically, this is accomplished by first drawing the span of which is a line through the origin (and, not coincidentally, the solution to ), and we translate, or push, this line along The translated line contains and is parallel to it is a translate of a line.
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.
If is consistent, the set of solutions to is obtained by taking one particular solution of and adding all solutions of
In particular, if is consistent, the solution set 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
so is another solution of On the other hand, if we start with any solution to then is a solution to since
See the interactive figures in the next subsection for visualizations of the key observation.
As in this important note, 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 (again not through the origin when the system is inhomogeneous), etc.
Again compare with this important note in Section 3.2.
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
The span of the columns of : this is the set of all such that is consistent.
Do not confuse these two geometric constructions! In the first the question is which ’s work for a given and in the second the question is which ’s work for some
