Systems of linear equations連立一次方程式れんりついちじほうていしき and augmented matrices拡大係数行列かくだいけいすうぎょうれつ

1Introduction

The essential point of this lecture is to convert a system of linear equations連立一次方程式れんりついちじほうていしき into a problem about a matrix行列ぎょうれつ containing only its coefficients係数けいすう.

Gaussian elimination掃はき出だし法ほう is a method for operating on the arrangement of coefficients, not on the written equations themselves. Therefore the correspondence between equations and matrices must be fixed first.

2Terms and definitions

A coefficient matrix係数行列けいすうぎょうれつ is the matrix行列ぎょうれつ formed by arranging the coefficients係数けいすう on the left-hand side of a system of linear equations連立一次方程式れんりついちじほうていしき.

An augmented matrix拡大係数行列かくだいけいすうぎょうれつ is the matrix obtained by adjoining the right-hand side右辺うへん column to the coefficient matrix係数行列けいすうぎょうれつ.

3Strategy

The goal is to put the system in the form

$Ax=b$Ax=b

Here $A$A is the coefficient matrix係数行列けいすうぎょうれつ, $x$x is the vector of unknowns未知数みちすう, and $b$b is the right-hand side右辺うへん vector.

4Intuitive explanation

If the variable symbols are written out every time, it becomes harder to see the essential operation. What actually changes is the list of coefficients係数けいすう and the right-hand side右辺うへん. So we extract only the coefficients and place them in a table.

The augmented matrix拡大係数行列かくだいけいすうぎょうれつ is notation for tracking the left-hand side coefficients and the right-hand side simultaneously.

5Precise explanation

The system of linear equations連立一次方程式れんりついちじほうていしき

can be written as

Thus

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

6Worked example

For

the coefficient matrix係数行列けいすうぎょうれつ is

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

There are three unknowns未知数みちすう and two equations, so $A$A is a $2\times 3$2\times3 matrix行列ぎょうれつ.

7Common misunderstandings

• The coefficient matrix係数行列けいすうぎょうれつ does not include the right-hand side右辺うへん. The matrix that includes the right-hand side is the augmented matrix拡大係数行列かくだいけいすうぎょうれつ.
• The order of the unknowns未知数みちすう must not be changed midway. The order of the columns represents the order of the unknowns.
• The number of equations and the number of unknowns need not be the same.

8Scope of validity

This notation works for linear equations. If nonlinear terms such as $xy$xy or $x^2$x^2 appear, the system cannot be represented in the same way by only a coefficient matrix係数行列けいすうぎょうれつ.

9Final forms

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}Ax=b$\boxed{Ax=b}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\left(A\mid b\right)\text{istheaugmentedmatrix}$\boxed{\left(A\mid b\right)\text{ is the augmented matrix}}

10Exercise links

11Related links