Row Reduced Echelon Form of a Matrix

DEFINITION 2.3.1 (Row Reduced Form of a Matrix)   A matrix $C$ is said to be in the row reduced form if

1. THE FIRST NON-ZERO ENTRY IN EACH ROW OF MATHEND000# IS MATHEND000#
2. THE COLUMN CONTAINING THIS MATHEND000# HAS ALL ITS OTHER ENTRIES ZERO.

A matrix in the row reduced form is also called a ROW REDUCED MATRIX.

EXAMPLE 2.3.2  

1. One of the most important examples of a row reduced matrix is the $n \times n$ identity matrix, $I_n.$ Recall that the $(i,j)^{\mbox{th}}$ entry of the identity matrix is
   $$I_{ij} = \delta_{ij} = \begin{cases} 1 & {\mbox{ if }} i = j \\ 0 & {\mbox{ if }} i \neq j. \end{cases}.$$
   $\delta_{ij}$ is usually referred to as the Kronecker delta function.
2. The matrices $\begin{bmatrix}0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}$ and $\begin{bmatrix}0 & 1 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$ are also in row reduced form.
3. The matrix $\begin{bmatrix}1 & 0 & 0 & 0 & 5 \\ 0 & 1 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$ is not in the row reduced form. (why?)

DEFINITION 2.3.3 (Leading Term, Leading Column)   For a row-reduced matrix, the first non-zero entry of any row is called a LEADING TERM. The columns containing the leading terms are called the LEADING COLUMNS.

DEFINITION 2.3.4 (Basic, Free Variables)   Consider the linear system $A {\mathbf x}= {\mathbf b}$ in $n$ variables and $m$ equations. Let $[C \; \; {\mathbf d}]$ be the row-reduced matrix obtained by applying the Gauss elimination method to the augmented matrix $[ A \; \; {\mathbf b}].$ Then the variables corresponding to the leading columns in the first $n$ columns of $[C \; \; {\mathbf d}]$ are called the BASIC variables. The variables which are not basic are called FREE variables.

The free variables are called so as they can be assigned arbitrary values and the value of the basic variables can then be written in terms of the free variables.

Observation: In Example 2.2.12, the solution set was given by

$$(x, y, z)^t = (1, 2-z, z)^t = (1, 2, 0)^t + z (0, -1, 1)^t, \;{\mbox{ with }} z \; {\mbox{ arbitrary}}.$$

That is, we had two basic variables, $x$ and $y,$ and $z$ as a free variable.

Remark 2.3.5   It is very important to observe that if there are $r$ non-zero rows in the row-reduced form of the matrix then there will be $r$ leading terms. That is, there will be $r$ leading columns. Therefore, IF THERE ARE MATHEND000# LEADING TERMS AND MATHEND000# VARIABLES, THEN THERE WILL BE MATHEND000# BASIC VARIABLES AND MATHEND000# FREE VARIABLES.