Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the lecture entitled "Describing Infinite Solution Sets Parametrically"
we saw how infinite solution sets could be described using parametric equations.
In this lecture, we will show that once an augmented matrix
representing a system of equations with infinitely many solutions
has been transformed to reduced row echelon form
it is especially easy to produce a parametric description of those solutions.
But before we describe the solutions of a system
we first need to know what type of solutions the system has.
Is it possible to tell if a system has a single unique solution
no solutions
or infinitely many solutions
by simply by looking at the matrix?
The answer is "yes"
once the matrix is in reduced row echelon form.
So let's see exactly how this is done.
Given a reduced row echelon matrix
the first step is to determine if the system is inconsistent, having no solutions.
This is easy, since if the matrix contains any row which has all zero coefficient entries
and a non-zero constant entry
then the system has no solutions.
This can often be determined during Gauss-Jordan elimination
even before the process is finished and the matrix is in reduced row echelon form.
Once we have eliminated the possibility that a system is inconsistent
we know that the system is consistent with one or more solutions.
The next step is to determine whether the system has a single unique solution
or infinitely many solutions.
We can determine this by looking at the positions of what are called "pivot columns".
So what exactly is a pivot column?
We know that in reduced row echelon matrices
the left-most non-zero entry in each row, called the row's "leading entry"
must be a "one".
The positions that these leading entries occupy in a matrix are called "pivot positions"
and the columns that contain these pivot positions are called "pivot columns".
The positions of these pivot columns
tells us what type of solutions the system has.
If every column to the left of the vertical line is a pivot column
then the system has a single unique solution.
And if any column left of the vertical line is not a pivot column
then the system has infinitely many solutions.
We have already seen that when a system has a single unique solution
it is easy to describe that solution
simply by looking at the reduced row echelon form matrix.
So how do we describe the solution set when there are infinitely many solutions?
In the lecture "Describing Infinite Solution Sets Parametrically"
we saw how to use parametric equations to describe an infinite solution set
when using the elimination method.
However, creating a parametric description from a reduced row echelon matrix
is an even simpler process.
In a reduced row echelon form matrix
the columns to the left of the vertical line each contain the coefficients for one variable.
But among these columns, the pivot columns and their associated variables
have a special significance.
When creating a set of parametric equations
the variables associated with the pivot columns
are treated as dependent variables
and the variables associated with the remaining columns are considered to be independent.
In the parametric equations, the values of the dependent variables
are expressed as functions of the independent variables which can vary freely.
For this reason, the independent variables are sometimes referred to as "free variables".
To create a parametric description of the system's solution set
we first convert the non-zero rows of the matrix into equations
and then rearrange the equations
so that the dependent variables are alone on the left side.
This gives us equations which express the dependent variables
as functions of the free variables.
But to express the solution set in parametric form
each variable in the system must be written as a function of "parameters"
which are a new set of independent variables created specifically to define the solution set.
The first step is to equate each free variable to a parameter.
So one parameter must be created for each free variable.
For example, in this system we can equate the free variable z
to a parameter which we will call "t".
The second step is to replace all the free variables in the remaining equations
by those parameters.
So in this example we replace z in all the other equations by t.
This gives us a set of parametric equations which express each system variable
as a function of a single parameter, t.
Creating this set of parametric equations was simple.
Since each dependent variable had a coefficient of one, and appeared in only one equation
it was a simple matter to rewrite those equations
so that each equation equated one dependent variable
to an expression containing only the free variable.
That free variable was then set equal to an independent parameter
which could then replace the free variable in the other equations.
This gave us a set of parametric equations expressing each system variable
as a function of a single parameter, t.
In this example, there was only one column left of the vertical line
that was not a pivot column
and so there was only one free variable.
Our solutions therefore contained only one parameter.
But now let's look at an example with more than one free variable
and thus more than one parameter.
In the lecture "Gauss-Jordan Elimination with Dependent Systems"
we solved a system of three linear equations which were all multiples of each other.
Once we expressed this system as a matrix
and transformed it to reduced row echelon form
we found that the matrix represented an equivalent system
consisting of a single equation in three variables.
So how can we write a set of parametric equations for this solution set?
Looking at the reduced row echelon matrix
we see that there is only one pivot position
and thus one pivot column.
Since this pivot column corresponds to the variable x
x is the dependent variable
and y and z are free variables.
Rewriting this equation so that the dependent variable x is alone on the left side
we then only need to equate the free variables y and z to parameters of our choosing.
Let's call these two parameters "s" and "t".
We can then write the first equation in terms of s and t instead of y and z.
The solution set of this system is now described by a group of parametric equations
which express each of the three variables as functions of two parameters s and t.
Now let's look at an example of a reduced row echelon matrix with three rows
but five columns to the left of the vertical line.
This matrix therefore represents a system of three linear equations in five variables.
Since this matrix contains three non-zero rows, it has three pivot positions
and thus three pivot columns.
As we have seen, the variables associated with these columns
will be the dependent variables in our parametric equations
and the remaining variables will be free variables.
Converting the three matrix rows into equations
and rearranging the equations
so that the dependent variables are alone on the left side
gives us equations which express the dependent variables x, z, and v
as functions of the free variables y and u.
To express the values of all five variables as functions of parameters
we simply equate the free variables y and u
to parameters which we will once again call s and t
and then express the other variables as functions of s and t instead of y and u.
The solution set of this system is now described by parametric equations
which express the values of each of the five variables
as functions of two parameters s and t.
In the first example, the system of equations contained three variables, x, y, and z
and so the system was graphically represented in three dimensions.
However, the solution set of the system corresponded to a 1-dimensional line.
Since the solution was 1-dimensional
its parametric representation required only a single parameter, t.
As this parameter t varies from negative infinity to positive infinity
each unique value of t produces a unique set of values for x, y, and z.
Each set of values is one solution of the system
corresponding to one point on the line.
In this example, the space containing the system and its solution set is 3-dimensional
because the system consists of equations in three variables x, y, and z.
All the possible combinations of values of those variables
correspond to all the possible points in this 3-dimensional space.
However, the solution set of that system is a subset of those values
whose coordinates correspond to points on a 1-dimensional line.
This line can be thought of as a 1-dimensional space
embedded within the system's 3-dimensional space.
A lower-dimensional space like this, embedded within a space of higher dimension
is called a "subspace".
So the solution set of the system
forms a subspace of lower dimension than the system itself
and the parameter t can be thought of as a single coordinate
which locates a point within this subspace.
The set of parametric equations
are therefore a way to translate the values of that coordinate t
into the values of the three coordinates x, y, and z
which locate the same point in the 3-dimensional space of the system of equations.
The system of equations in the second example also contained three variables
and so the graph of this system also required three dimensions.
However, the solution set of this system formed a 2-dimensional plane.
The parametric representation of the solution set
therefore required two parameters which we called s and t.
As s and t each independently vary from negative infinity to positive infinity
each unique combination of values produces a unique set of values for x, y, and z.
Each set of values is one solution of the system
corresponding to one point on the plane.
Although the space containing this system is 3-dimensional
since the system consists of equations in three variables x, y, and z
the solution set of that system is a 2-dimensional subspace
embedded within that 3-dimensional space.
The parameters s and t can therefore be thought of as coordinates
which locate a point within this 2-dimensional subspace
and the parametric equations can be thought of as a way
to translate the values of the two parameters s and t
into the values of the three coordinates x, y, and z
which locate the same point in the 3-dimensional space of the system of equations.
In the third example, the system of equations contained five variables
and so graphing this system would require five dimensions
not a very practical task in this universe.
However, because the system had two free variables
the parametric representation of the solution set
still required only two parameters which we once again called s and t.
The solution set of that system can therefore be thought of
as a 2-dimensional subspace embedded within a 5-dimensional space
and the parameters s and t can be thought of as coordinates
which locate a point within the 2-dimensional subspace.
So even though it is impractical to visualize this system graphically
the parametric equations are a way to translate the values of the two parameters s and t
into values of the five coordinates x, y, z, u, and v
which locate the same point in the 5-dimensional space of the system of equations.
So we have seen that in a reduced row echelon form matrix
the pivot columns divide the system variables into dependent variables
and free variables.
The number of free variables corresponds to the number of dimensions required
to graphically represent the solution set.
If there are two free variables
the infinite solution set will be graphically represented by a 2-dimensional plane.
If there is one free variable
the infinite solution set will be represented by a 1-dimensional line.
And if all columns left of the vertical line are pivot columns
there are no free variables
and the solution set will be a zero-dimensional point
corresponding to a single unique solution.
In the next lecture, we will use Gauss-Jordan elimination
to transform systems of equations with no solutions
and follow how its graphical representation changes during each step of the process.