2005/7 Linear system-1

The Linear Equation System and Eliminations

2005/7 Linear system-2

Linear equation system over F

Coefficients a1, a2, a3, …, anF and constant term bF. a

1 is called a leading coefficient ( 領先係數 ) and x1 is calle

d a leading variable.注意：

(1) 線性方程式之變數不可以是相乘或是開根號，且 變數不能被包含在三角、指數或對數函數裡面。 (2) 變數只能以第一冪次的方程式表示 。

2005/7 Linear system-3

Ex: Linear or Non-linear723 )( yxa 2

21

)( zyxb

0102 )( 4321 xxxxc 221 4)

2sin( )( exxd

2 )( zxye 42 )( yef x

032sin )( 321 xxxg 411

)( yx

h( )非第一冪次變數不能相乘

指數

三角函數 非第一冪次

線性

線性

線性

線性

非線性

非線性

非線性

非線性

2005/7 Linear system-4

n

mnmm

n

n

AAA

aaa

aaa

aaa

A

21

21

22221

11211

nx

x

x

2

1

x

12211

2222121

1212111

mnmnmm

nn

nn

xaxaxa

xaxaxa

xaxaxa

A

x

Linear combination of the column vectors of matrix A

mn

n

n

n

a

a

a

x

2

1

1

21

11

1

ma

a

a

x

2

22

21

2

ma

a

a

x

Ax = x1A1 + x2A2 + + xnAn

2005/7 Linear system-5

The Solution of a Linear Equation System

For a linear equation system, only one of the following

statements will be true:

(1) There is exactly one solution. (consistent)

(2) There are infinitely many solutions. (consistent)

(3) There is no solution. (inconsistent)

2005/7 Linear system-6

Example:

(1)

(2)

(3)

13

yxyx

6223

yxyx

13

yxyx

)(唯一解

)(無限多組解

)(無解

兩相交直線

兩重疊直線

兩平行直線

2005/7 Linear system-7

Ex: Use back substitution to solve the linear equation system

(2)(1)

252

yyx

Sol: Let substitute to (1)2y

15)2(2

xx

2 ,1 yxThe only solution is

2005/7 Linear system-8

Equivalent

If two linear equation systems have the exactly same solution sets, then we say they are equivalent to each other.

The following operations will produce equivalent linear equation systems.

(1) Exchange two equations.

(2) Multiple a nonzero constant to an equation.

(3) Add two equations.

2005/7 Linear system-9

Ex: Solve the linear equation system.

(3)(2)(1)

17552

43932

zyxyx

zyx

Sol:

(4) 17552

53932

(2)(2)(1)

zyxzyzyx

(5)

153932

(3)(3)2)((1)

zyzyzyx

2005/7 Linear system-10

The solution is 2 ,1 ,1 zyx

(6)

4253932

(5)(5)(4)

zzyzyx

253932

)6((6) 21

zzyzyx

2005/7 Linear system-11

Ex: Solve the given system of linear equations.

(3)(2)(1)

13222213

321

321

321

xxxxxxxxx

Sol:

)5()4(

24504513

(3)(3))1((1)

(2)(2)2)((1)

32

32

321

xxxxxxx

2004513

)5()5()1()4(

32

321

xxxxx

contradictionThis system has no solution.

2005/7 Linear system-12

A system of Linear Equations (or a linear equation system) over a field (real numbers R or complex number C)

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

2211

22222121

11212111

mnmnmm

n

n

b

b

b

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

(matrix of coefficients) A x b

bx A1 nnm 1m

2005/7 Linear system-13

The augmented matrix ( 增廣矩陣 )

The coefficient matrix ( 係數矩陣 )

][ 3

2

1

321

3333231

2232221

1131211

bA

b

bbb

aaaa

aaaaaaaaaaaa

mmnmmm

n

n

n

A

aaaa

aaaaaaaaaaaa

mnmmm

n

n

n

321

3333231

2232221

1131211

2005/7 Linear system-14

Three elementary row operations

jiij RRr :(1) 交換兩列

iiki RRkr )(:)(

(2) 乘上一個非零常數到某列

jjikij RRRkr )(:)((3) 一列的倍數加到另一列

row equivalent若一矩陣可由另一矩陣的一些基本列運算來獲得，則此兩個矩陣稱為列等價 (row equivalent)

2005/7 Linear system-15

Ex: Elementary row operations

143243103021

143230214310

12r

212503311321

212503312642 )(

121

r

8133012303421

251212303421 )2(

13r

2005/7 Linear system-16

The row-echelon form ( 列梯形形式 )(1) 全部為零的列在矩陣最底下(2) 不全為零的列，其第一個非零元素為 1 ，稱為領先1 (leading 1)

(3) 對兩相鄰的非零列而言，較高列之領先 1 出現在較低列之領先 1 的左邊

The reduced row-echelon form ( 列簡梯形形式 )(1) ~ (3) 同上

(4) 在領先 1 的那一行除了領先 1 以外的位置全部為零

2005/7 Linear system-17

列簡梯形形式

列簡梯形形式列梯形形式

列梯形形式

Ex: 判斷下列矩陣為列梯形形式或列簡梯形形式

210030104121

10000410002310031251

000031005010

0000310020101001

310011204321

421000002121

2005/7 Linear system-18

The Gaussian elimination ( 高斯消去法 ) 將矩陣化簡為列梯形形式的程序

The Gauss-Jordan elimination ( 高斯 - 喬登消去法 )

將矩陣化簡為列簡梯形形式的程序 注意： (1) 每個矩陣只有一個列簡梯形形式 (2) 每個矩陣可以有很多種列梯形形式 ( 不同的列運算 會產生不同的列梯形形式 )

2005/7 Linear system-19

最左邊的非零行

產生 leading 1

讓在 leading 1 下的元素為 0

leading 1

產生 leading 1

最左邊的非零行

Ex: 高斯消去法與高斯喬登消去法之步驟說明

1565422812610421270200

1565421270200281261042

12r

15654212702001463521

)(1

21

r

2917050012702001463521)2(

13r

子矩陣

2005/7 Linear system-20

讓在 leading 1 下的元素為 0

讓 leading 1 以外的其他位置為 0

leading 1

最左邊的非零行產生 leading 1

leading 1

291705006270100

1463521)(2

21r

12100006270100

1463521)5(23

r

2100006270100

1463521)2(3r

)(列梯形形式

210000100100703021)6(

31r )(

3227

r )5(21r

)(列簡梯形形式

子矩陣

2005/7 Linear system-21

Ex: Use Gauss-Jordan elimination to solve the system of linear equations.

1755243932

zyxyx

zyx

Sol:

增廣矩陣

1755240319321

111053109321)2(

131

12 , rr

2100531093211

23r

列簡梯形形式

210010101001)9(

31)3(

322

21 , , rrr

列梯形形式

211

zy

x

2005/7 Linear system-22

Ex: Solve the linear equation system.

1 530242

21

311

xxxxx

Sol:

10530242

增廣矩陣

13102501

列簡梯形形式)2(

21)1(

2)3(

12)(

1 ,,,21 rrrr

相對的線性方程式系統

13 25

32

31

xxxx

3

21

) variablefree(

, ) variableleading(

x

xx

：自由變數：領先變數

txx

txx

3131

5252

32

31

Let tx 3

This system has infinitely many solutions.

2005/7 Linear system-23

The homogeneous system 若一線性方程系統的常數項均為零時， 則此系統為齊次系統

0

0 0 0

332211

3333232131

2323222121

1313212111

nmnmmm

nn

nn

nn

xaxaxaxa

xaxaxaxaxaxaxaxaxaxaxaxa

2005/7 Linear system-24

Trivial solution of a homogeneous system ( 顯然解 )

Nontrivial solution( 非顯然解 )

顯然解之外的其他解

0321 nxxxx

注意： (1) 所有的齊次系統均為一致性 (consistent) 系統 (2) 若系統的方程式比變數少，則有無限多組解 (3) 對於一個齊次系統來說，下列有一為真

(a) 系統只有一個顯然解 (b) 系統除了顯然解外還有無限多組解

( 任意 n變數齊次系統的解 )

2005/7 Linear system-25

Ex: Find the solution of the given homogeneous system.

032

0

321

321

xxx

xxx

Sol:

03120311

增廣矩陣

01100201

列簡梯形形式)1(

21)(

2)2(

12 ,, 31

rrr

3

21

,

x

xx

自由變數：領先變數：

Let tx 3

Rttxtxtx , , ,2 321

solution) (trivial 0,0 321 xxxt