Tutorials on Advanced Math and Computer Science Concepts

Reduced Row Echelon Form and Rank

We can extend the idea of row echelon format to make it easier to identify the solution to the system. This new format is called reduced row echelon form and has the following properties.

1. The matrix is in row echelon form
2. All leading entries are 1
3. In a column with a leading 1, all other entries are zeros.

The value of this format is that once we isolate for a single coefficient, we immediately know its value since the coefficient has been set to 1, due to the second property. This makes it easier to substitute the value into previous equations as well since they will also have a leading 1 coefficient. Further to this, there are new properties that we can derive with reduced row echelon form, which will give us valuable information about the system we are working with.

There are a few different types of linear systems that we may encounter. Generally, there are consistent systems, which have at least one solution, and inconsistent systems, which have no solutions. In addition to this, a consistent system could potentially have more than one solution. We can determine what type of system we have using the leading coefficients of it.

Theorem: Suppose that we have an augmented matrix $\left[\begin{array}{cc} A & | & \vec{b} \end{array}\right]$, that represents a system of linear equations. If this augmented matrix is in row echelon form:

1. The given system is inconsistent if and only if some row is of the form $\left[\begin{array}{cc} 0 & 0 & \dots & 0 & | & c \end{array}\right]$, where $c \ne 0$
2. If the system is consistent, it has a unique solution if the number of variables in the system is equal to the number of leading coefficients. If the number of leading coefficients is less than the number of variables, there are many solutions.

The first statement is true because the equation $0 = c$ has no solution, given that we specify $c \ne 0$. Since a row of the system has no solution, there is no way to find a solution for the whole system, meaning it is inconsistent. The second statement is true because if we have exactly one leading coefficient for each variable, we have exactly one solution for each equation. The alternative would mean that one of the variables is a free variable and therefore can take on any value, giving us a non-unique solution set.

From this theorem, we can see that knowing the number of leading coefficients can tell us a lot about a system of equations. Reduced row echelon form allows us to further extend this using the idea of a rank of a matrix.

The rank of the matrix is defined as the number of leading 1s the matrix has in reduced row echelon form. The reason that we need to use reduced row echelon form is row echelon form is not necessarily unique, whereas reduced row echelon form is. This just means it is easier and more reliable to work with reduced row echelon form.

We can alter the previous theory to discuss rank to describe linear systems.

Theorem: Let $\left[\begin{array}{cc} A & | & \vec{b} \end{array}\right]$ by a system of m linear equations with n variables. The following is true:

1. The system is consistent if and only if the rank of A is equal to the rank of $\left[\begin{array}{cc} A & | & \vec{b} \end{array}\right]$
2. If the system is consistent, the number of parameters in the general solution is n - rank(A), the number of variables minus the rank of the matrix A.
3. The system is consistent for all $\vec{b} \in \mathbb{R}^n$ if and only if rank(A) = m

Using these theories, we can easily conclude if a system is consistent or not, meaning we will know if we can expect a solution to it.