There is one possibility for the row reduced form of a matrix that we did not see in Section 2.2.
Consider the linear system
We solve it using row reduction:
This row reduced matrix corresponds to the linear system
In what sense is the system solved? We rewrite as
For any value of
there is exactly one value of
that make the equations true. But we are free to choose any value of
We have found all solutions: it is the set of all values
This is called the parametric form for the solution to the linear system. The variable
is called a free variable.
Figure 2 A picture of the solution set (the yellow line) of the linear system in this example. There is a unique solution for every value of
move the slider to change
Given the parametric form for the solution to a linear system, we can obtain specific solutions by replacing the free variables with any specific real numbers. For instance, setting
in the last example gives the solution
gives the solution
Consider a consistent system of equations in the variables
be a row echelon form of the augmented matrix for this system.
is a free variable if its corresponding column in
In the above example, the variable
was free because the reduced row echelon form matrix was
the free variables are
(The augmentation column is not free because it does not correspond to a variable.)
The parametric form of the solution set of a consistent system of linear equations is obtained as follows.
Moving the free variables to the right hand side of the equations amounts to solving for the non-free variables (the ones that come pivot columns) in terms of the free variables. One can think of the free variables as being independent variables, and the non-free variables being dependent.
The solution set of the system of linear equations
as we saw in this example. These equations are called the implicit equations for the line: the line is defined implicitly as the simultaneous solutions to those two equations.
The parametric form
can be written as follows:
This called a parameterized equation for the same line. It is an expression that produces all points of the line in terms of one parameter,
One should think of a system of equations as being an implicit equation for its solution set, and of the parametric form as being the parameterized equation for the same set. The parametric form is much more explicit: it gives a concrete recipe for producing all solutions.
While you can certainly write parametric solutions in point notation, it turns out that vector notation is ideally suited to writing down parametric forms of solutions.
It is sometimes useful to introduce new letters for the parameters. For instance, instead of writing