Row echelon form階段形かいだんけい and reduced row echelon form簡約階段形かんやくかいだんけい

1Introduction

The important point of this lecture is that row echelon form階段形かいだんけい is a shape that organizes a matrix行列ぎょうれつ so that solutions can be read, and that pivots主成分しゅせいぶん indicate the essential variables and the rank階数かいすう.

In Gaussian elimination掃はき出だし法ほう, we do not merely make matrix entries smaller. We organize the positions of the pivots主成分しゅせいぶん step by step. The organized shape is row echelon form階段形かいだんけい.

2Terms and definitions

A pivot主成分しゅせいぶん is the first nonzero entry from the left in each nonzero row.

A matrix is in row echelon form行階段形ぎょうかいだんけい when the pivots move to the right as one goes down the rows and all zero rows are gathered at the bottom.

A matrix is in reduced row echelon form簡約行階段形かんやくぎょうかいだんけい when, in addition, each pivot is 1 and every other entry in the pivot column is 0.

3Strategy

Use elementary row operations行基本変形ぎょうきほんへんけい to eliminate nonzero entries below pivot positions and arrange pivots in a staircase pattern. If the solution must be read completely, reduce further by making the entries above and below each pivot column主成分列しゅせいぶんれつ zero.

4Intuitive explanation

In row echelon form階段形かいだんけい, the leading variables appear from left to right. A column with a pivot is a leading-variable column, and a column without a pivot becomes a free-variable自由変数じゆうへんすう column.

Thus row echelon form階段形かいだんけい is both an intermediate computational shape and a table for checking the degrees of freedom in the solution.

From the viewpoint of a linear transformation線型変換せんけいへんかん, a pivot marks an input direction that remains independently effective on the output side. A column without a pivot is a direction that can be explained by combinations of pivot columns, and freedom remains there.

Therefore row echelon form階段形かいだんけい is not merely spreadsheet tidying. It is a form for reading which directions a map写像しゃぞう keeps independent独立どくりつ and which directions become dependent従属じゅうぞく on other directions.

5Precise explanation

For example,

is in row echelon form行階段形ぎょうかいだんけい. The pivots主成分しゅせいぶん are in columns 1 and 2, so the number of pivots is 2. Therefore the rank階数かいすう is 2.

In reduced row echelon form簡約行階段形かんやくぎょうかいだんけい, the other entries in each pivot column主成分列しゅせいぶんれつ are also made 0. For example,

is in reduced row echelon form簡約行階段形かんやくぎょうかいだんけい.

6Worked example and consistency check

The augmented matrix拡大係数行列かくだいけいすうぎょうれつ

is in row echelon form階段形かいだんけい. The second row gives $y=1$y=1. Substituting into the first row gives $x+2=5$x+2=5, hence $x=3$x=3. Row echelon form階段形かいだんけい is a shape for recovering the solution from bottom to top.

For an augmented matrix拡大係数行列かくだいけいすうぎょうれつ, pay attention to the case where a pivot appears only in the last column. For example,

contains the row $0x+0y=1$0x+0y=1. This says $0=1$0=1, a contradiction, so no solution exists. In terms of rank階数かいすう, the coefficient matrix係数行列けいすうぎょうれつ has rank 1, but the augmented matrix拡大係数行列かくだいけいすうぎょうれつ has rank 2, so

This mismatch means that the right-hand side右辺うへん $b$b is not in the column space列空間れつくうかん.

7Common misunderstandings

• A pivot is not just any nonzero entry. It is the first nonzero entry from the left in each nonzero row.
• Row echelon form階段形かいだんけい and reduced row echelon form簡約階段形かんやくかいだんけい are not the same. In the reduced form, the other entries in pivot columns are also 0.
• The rank階数かいすう is not the number of nonzero entries; it is the number of pivots.

8Scope of validity

A row echelon form階段形かいだんけい obtained by elementary row operations行基本変形ぎょうきほんへんけい is not necessarily unique. However, the reduced row echelon form簡約行階段形かんやくぎょうかいだんけい is unique. The rank階数かいすう and the number of pivots do not depend on the choices made during the operations.

9Final forms

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\text{rank}=\text{numberofpivots}$\boxed{\text{rank}=\text{number of pivots}}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\text{columnswithoutpivotsgivefreevariables}$\boxed{\text{columns without pivots give free variables}}

10Exercise links

11Related links