Echelon Form

A System Of Equations is in echelon form (EF) if the following 3 conditions hold:

1. All rows of zero are at the bottom
2. The first non-zero entry in a row is 1. This is the leading one
3. Each leading one is to the right of all leading ones above it Note: Condition 1 causes the leading terms to form a staircase from top-left to bottom right

Formal Definition

1. In each equation either all the coefficients are zero or the first non-zero coefficients is 1. We say that the firs tnon-zero term $1x_{j(i)}$ is the leading term of its row its column is $j_i$
2. Each $r\ne i$, the coefficient $a_{rj(i)}$ of $x_{ji}$ is zero (Only non-zero entry in the column)
3. $j(i)<j(i+1)$ whenever rows $i$ and $i+1$ have non-zero entites (“Echelon/staircase pattern”)

Examples

Concepts

• Every Linear System Has A Row-Echelon Form Theorem
• Elimination Algorithm
• Basic and Free Variables