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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