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 Gaussian elimination掃はき出だし法ほう

1Introduction

The essential point of this lecture is that Gaussian elimination掃はき出だし法ほう is not just a trick for eliminating coefficients. It transforms 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 an equivalent同値どうち form and makes the structure構造こうぞう of the solution explicit.

If one relies only on substitution or ad hoc elimination, the structure becomes unclear when there are many equations. Gaussian elimination掃はき出だし法ほう systematizes that procedure.

2Terms and definitions

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 is a collection of equations in which the unknowns未知数みちすう appear linearly.

An elementary row operation行基本変形ぎょうきほんへんけい is one of the following operations on rows: swapping rows, multiplying a row by a nonzero constant, or adding a constant multiple of one row to another row.

3Strategy

First convert the system to an augmented matrix拡大係数行列かくだいけいすうぎょうれつ. Then use elementary row operations行基本変形ぎょうきほんへんけい to secure pivots主成分しゅせいぶん from left to right and eliminate entries below or above them.

4Intuitive explanation

Gaussian elimination掃はき出だし法ほう amounts to choosing a representative entry for a column in a table and clearing entries above or below it. Moving from left to right reveals which variables are determined and which variables remain free.

5Precise explanation

5.11. Augmented matrix拡大係数行列かくだいけいすうぎょうれつ

For example,

is represented by

5.22. Elementary row operation行基本変形ぎょうきほんへんけい

Apply

This subtracts twice the first row from the second row and gives

From the second row, $-3y=-3$-3y=-3, so $y=1$y=1. Substituting into the first row gives $x+1=2$x+1=2, hence $x=1$x=1. Therefore the solution is $x=1,y=1$x=1,y=1.

5.33. What can be decided

After elimination, a contradictory row means there is no solution, and a remaining free variable自由変数じゆうへんすう means there are infinitely many solutions.

The three smallest cases make the checkpoints explicit.

In the unique-solution case, each unknown未知数みちすう is determined by a pivot. In the no-solution case, a row appears whose coefficient side is all zero but whose right-hand side右辺うへん is nonzero. In the infinitely-many-solutions case, a free variable自由変数じゆうへんすう can be moved arbitrarily, so the solution set retains a degree of freedom.

6Criteria

• If there are several unknowns未知数みちすう and several equations, prefer Gaussian elimination掃はき出だし法ほう.
• The elimination result is also useful when deciding whether the solution is unique or whether no solution exists.
• At every step, check that the system is being replaced by an equivalent同値どうち system.
• To decide the solution type, check for contradictory rows, count the pivots, and then check whether free variables自由変数じゆうへんすう remain.

7Final form

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\text{systemoflinearequations}\to \text{augmentedmatrix}\to \text{Gaussianelimination}$\boxed{\text{system of linear equations} \rightarrow \text{augmented matrix} \rightarrow \text{Gaussian elimination}}

8In one sentence

• Gaussian elimination掃はき出だし法ほう is both a procedure for solving 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 and a method for checking the structure構造こうぞう of its solutions.

9Exercise links