Systems of Linear Equations: Row Echelon Form, Free Variables, and Gaussian Elimination

In mathematics, linear algebra deals with various mathematical concepts and techniques used to manipulate vector spaces, particularly Euclidean space. One essential aspect of linear algebra is solving systems of linear equations, which involve finding values for a set of variables that satisfy several equations simultaneously. By employing methods like Gaussian elimination and representing matrices in specific forms, such as the row echelon form (REF) and reduced row echelon form (RREF), we can effectively solve and analyze these systems. Let's delve into the details of these concepts.

Row Echelon Form (REF)

The row echelon form (REF) is a special arrangement of the rows in a matrix. It is obtained from the augmented matrix representation of a system of linear equations through a series of elementary row operations as follows:

1. Swapping any two rows.
2. Multiplying a row by a non-zero constant.
3. Adding a multiple of one row to another row.

The REF has the following properties:

• All elements below the leading diagonal elements in each column are zero.
• Each leading diagonal element is non-zero.
• For any element just above the leading diagonal element in a column, there is a column of all zeros until the leading diagonal element.

For instance, consider the matrix ([A|B]) and the corresponding augmented matrix (\begin{bmatrix}A&B\\mathbf{I}&O\end{bmatrix}), where (A) is the coefficient matrix, (B) is the constant matrix, and (\mathbf{I}) is the identity matrix. The REF of this matrix would look something like (\begin{bmatrix}R_{1}|N_{1}\ R_{2}|N_{2}\end{bmatrix}), where (R_{1}) and (R_{2}) are the first two rows, and (N_{1}) and (N_{2}) are the first two columns of the identity matrix.

Free Variables

In a system of linear equations, the free variables are those variables whose values can be chosen freely without violating the consistency of the system. In other words, if the constants in a row are linearly dependent on the variables in the corresponding column, then those variables are free variables.

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations by transforming them into an equivalent system through various elementary row operations. It involves performing a series of operations to convert the augmented matrix into one with specific properties:

• Upper triangular matrix: All the elements below the diagonal are zero.
• Leading diagonal non-zero elements: The first non-zero element in any row appears to the right of the first non-zero element above it.

This process can be summarized into three main steps:

Step 1: Choose a variable and solve one equation for it.

For example, if we have two equations (x + y = 2) and (2x + y = 3), we can choose (x) and eliminate it from the second equation by subtracting the first equation from the second. This yields (-(1)(x + y = 2) + 2x + y = 3), which simplifies to (x = 1). Now that we know the value of (x), we can substitute this back into either equation to get rid of (x) and find the corresponding value of (y).

Step 2: Eliminate variables from the remaining equations.

Using Gaussian elimination, we transform the system into another equivalent system, effectively eliminating one variable. For instance, consider the following three equations: (x + y = 2), (-2x + y = 1), and (3x + 4y = 7). By subtracting the first equation from the second, we get (x = -\frac{1}{2}). Then, substituting this back into either the second or third equation, we find that (y = \frac{3}{2}). Now, the system has been reduced to a single equation, which allows for easy solution.

Step 3: Solve the final reduced system.

In the case of two equations, such as (x + y = 2) and (2x + y = 3), the process may lead to inconsistent systems with no solutions. In this scenario, we need to determine whether it is possible to adjust the constants on the right side of the system so that it becomes consistent. If there exists a set of values for the constants that makes the system consistent, then the original system also has the same number of solutions. However, if the system remains inconsistent, then it does not have any solutions.