Chapter1.2 Row reduction and Echelon forms
In this post, we’ll study the following
- A nonzero row of column
- A leading entry of row
- Echelon form
- Reduced echelon form
- Uniqueness of the Reduced Echelon Form
- Row reduction algorithm
- Solution of linear systems
- Existence and Uniqueness Theorem
The key point of this post is that we can get a reduced echelon form with row reduction.
And if you make a linear system a reduced echelon form, it’s easy to find its solution.
A nonzero row or column
A nonzero row or column means that there is at least one item in the row or column that is not zero.
A leading entry of row
A leading entry of a row means the leftmost nonzero entry in the row.
Echelon form
Echelon form satisfies two conditions.
- All nonzero rows are above all zeros rows.
- The leading entry in a row is in the column to the right of the leading entry above it.
If you’re just looking at the text, it might not make sense. The following illustration should help you understand.
Reduced echelon form
A reduced row echelon form must satisfy two additional conditions beyond the basic two conditions of a echelon form.
(3) The leading entry in a nonzero row must be 1.
(4) Except for the part where the leading entry is 1, the rest of the column must be all zeros.
And the position of 1 is called the pivot position.
Uniqueness of the Reduced Echelon Form
For each matrix, there is only one reduced row echelon form.
There are no other forms of reduced row echelon form.
Row reduction algorithm
Now, let’s examine the process of creating a reduced row echelon form.
STEP 1. Start with the leftmost nonzero column. This is the pivot column, and its location is the pivot position.
In this case, since the pivot position is 0, an interchange with another row is necessary.
STEP 2. Select a nonzero entry in the pivot column as a pivot. if necessary, interchange rows to move this entry into pivot position.
STEP 3. Use row replacement operations to create zeros in all positions below the pivot.
STEP 4. Apply steps 1–3 to the submatrix.
The echelon form has been created through steps 1–4, known as the forward phase.
Now, let’s create the reduced echelon form through step 5, known as the backward phase.
STEP 5. Make the bottom pivot 1 and turn everything above the pivot into 0s.
In this way, the reduced echelon form can be created through the fifth step.
Solution of linear systems
If an augmented matrix is transformed into a reduced echelon form using the row reduction algorithm, it becomes easier to find the solution to a system of linear equations.
Let’s take a look at the augmented matrix in its reduced echelon form, obtained through row reduction.
This can be represented as a system of linear equations.
From the above system of linear equations, the solution can be found as follows.
It is important to clearly understand the terms general solution, basic variables, and free variables.
(1) Free variable
- x_3 is free variable.
- This is because x_3 can take any value and still satisfy the equation 0 = 0.
- A Free Variable is assigned to a column that does not have a pivot position. You can check at <pic 1>. Row 3 doesn’t have pivot position.
(2) Basic variables
- Basic variables refer to those variables that are expressed in terms of the free variables.
(3) General solution
- The general solution refers to the solution that is expressed in terms of both basic and free variables.
Existence and Uniqueness Theorem
(1) If a system of linear equations is consistent, then in its augmented matrix, the constant term column ‘b’ does not have a pivot position.
This means there should not be a row that is zero except for the ‘b’ term.
For example, the equation 0x_1 + 0x_2 = 3. It represents an impossible equation
(2) If a linear system is consistent, then
(i) if there are no free variables, there is exactly one solution.
(ii) if there is one or more free variables, there are infinitely many solutions.
