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Chapter 1Systems of Linear Equations
感謝 大葉大學 資訊工程系黃鈴玲老師 提供
Linear Algebra
Ch1_2
Definition• An equation such as x+3y=9 is called a linear equation.
( 線性方程式 )
• The graph of this equation is a straight line in the x-y plane.
• A pair of values of x and y that satisfy the equation is called a solution.
1.1 Matrices and Systems of Linear Equations
system of linear equations ( 線性聯立方程式 )
Ch1_3
Definition A linear equation in n variables ( 變數 ) x1, x2, x3, …, xn has
the form
a1 x1 + a2 x2 + a3 x3 + … + an xn = b
where the coefficients ( 係數 ) a1, a2, a3, …, an and b are real numbers ( 實數 ).
常見數系的英文名稱: natural number ( 自然數 ), integer ( 整數 ), rational number ( 有理數 ), real number ( 實數 ), complex number ( 複數 ) positive ( 正 ), negative ( 負 )
Ch1_4
Figure 1.2No solution ( 無解 )–2x + y = 3–4x + 2y = 2 Lines are parallel.No point of intersection. No solutions.
Solutions for system of linear equations
Figure 1.1Unique solution ( 唯一解 ) x + 3y = 9–2x + y = –4 Lines intersect at (3, 2)Unique solution: x = 3, y = 2.
Figure 1.3Many solution ( 無限多解 )4x – 2y = 66x – 3y = 9 Both equations have the same graph. Any point on the graph is a solution. Many solutions.
Ch1_5
A linear equation in three variables corresponds to a plane in three-dimensional ( 三維 ) space.
Unique solution
※ Systems of three linear equations in three variables:
Ch1_6
No solutions
Many solutions
Ch1_7
How to solve a system of linear equations?
Gauss-Jordan elimination. ( 高斯 - 喬登消去法 )
1.2 節會介紹
Ch1_8
Definition• A matrix ( 矩陣 ) is a rectangular array of numbers. • The numbers in the array are called the elements ( 元素 ) of
the matrix.
Matrices
1298
520
653
C
38
50
17
B 157
432A
注意矩陣左右兩邊是中括號不是直線,直線表示的是行列式。
Ch1_9
Submatrix (子矩陣 )
A matrix
215
032
471
A
Row (列 ) and Column (行 )
3column 2column 1column 2 row 1 row
1
4
5
3
7
2 157 432
157
432
A
A of ssubmatrice
25
41
1
3
7
15
32
71
RQP
Ch1_10
Identity Matrices (單位矩陣 ) diagonal ( 對角線 ) 上都是 1 ,其餘都是 0 , I 的下標表示 size
100010001
1001
32 II
Location
7 ,4 157
4322113
aaAaij 表示在 row i, column j 的元素值也寫成 location (1,3) = 4
Size and Type
matrixcolumn a matrix row a matrix square a
matrix 13 matrix 41 matrix 33 32 :Size
2
3
8
5834
853
109
752
542
301
Ch1_11
matrix of coefficient and augmented matrix
62
3 32
2
321
321
321
xxx
xxx
xxx
Relations between system of linear equations and matrices
係數矩陣 擴大矩陣
隨堂作業: 5(f)
tcoefficien ofmatrix
211
132
111
matrix augmented
6211
3132
2111
Ch1_12
給定聯立方程式後,不會改變解的一些轉換
Elementary Transformation
1. Interchange two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a multiple of one equation to another equation.
將左邊的轉換對應到矩陣上Elementary Row Operation ( 基本列運算 )
1. Interchange two rows of a matrix. ( 兩列交換 )
2. Multiply the elements of a row by a nonzero constant. ( 某列的元素同乘一非零常數 )
3. Add a multiple of the elements of one row to the corresponding elements of another row. ( 將一個列的倍數加進另一列裡 )
Elementary Row Operations of Matrices
Ch1_13
Example 1Solving the following system of linear equation.
623322
321
321
321
xxxxxxxxx
621131322111
623322
321
321
321
xxxxxxxxx
SolutionEquation MethodInitial system:
Analogous Matrix MethodAugmented matrix:
1
2
32
321
xx
xxx
Eq2+(–2)Eq1
Eq3+(–1)Eq1
832011102111
R2+(–2)R1R3+(–1)R1
符號表示 row equivalent
832 32 xx
Ch1_14
1051
32
3
32
31
xxxxx
0150011103201
21
32
3
32
31
xxxxx
210011103201
21
1
3
2
1
xxx
2100
1010
1001
Eq1+(–1)Eq2
Eq3+(2)Eq2
(–1/5)Eq3
Eq1+(–2)Eq3
Eq2+Eq3
The solution is .2 ,1 ,1 321 xxx The solution is
.2 ,1 ,1 321 xxx
8321
2
32
32
321
xxxxxxx
832011102111
R1+(–1)R2 R3+(2)R2
(–1/5)R3
R1+(–2)R3 R2+R3
隨堂作業: 7(d)
Ch1_15
Example 2Solving the following system of linear equation.
83318521242
321
321
321
xxxxxxxxx
Solution (請先自行練習 )
83311851212421
41106330
12421
R23
1
41102110
12421
620021108201
3100
2110
8201
R1R3
2)R1(R2
R2)1(R3
(2)R2R1
R32
1
3100
1010
2001
R3R2
2)R3(R1
.
3
1
2
solution
3
2
1
x
x
x
Ch1_16
Example 3Solve the system
7 2
328 63
441284
21
321
321
xx
xxx
xxx
7012
32863
441284
7012
32863
11321
15630
1100
11321
1100
5210
11321
1100
5210
1101
.
1100
3010
2001
.1 ,3 ,2 issolution The 321 xxx
R23
1
R12R3
3)R1(R2
R3R2
1100
15630
11321
R14
1
2)R2(R1
2R3R2
1)R3(R1
Solution (請先自行練習 )
隨堂作業: 10(d)(f)
Ch1_17
Summary
7012
32863
441284
]:[ BA
A BUse row operations to [A: B] :
.
1100
3010
2001
7012
32863
441284]:[]:[ XIBA ni.e.,
Def. [In : X] is called the reduced echelon form ( 簡化梯式 ) of [A : B].Note. 1. If A is the matrix of coefficients of a system of n equations
in n variables that has a unique solution, then A is row equivalent to In (A In).
2. If A In, then the system has unique solution.
7 2
328 63
441284
21
321
321
xx
xxx
xxx
Ch1_18
Example 4 Many SystemsSolving the following three systems of linear equation, all of which have the same matrix of coefficients.
3321
2321
1321
42
for 42
3
bxxx
bxxx
bxxx
in turn 433
,210
,11118
3
2
1
bbb
Solution
42114213111412308311
.
2
1
2
,
1
3
0
,
2
1
1
3
2
1
3
2
1
3
2
1
x
x
x
x
x
x
x
x
x
123110315210308311
212100
315210
013101
212100
131010
201001
R2+(–2)R1 R3+R1
1)R2(R3
R2R1
R32R2
R3)1(R1
隨堂作業: 13(b)The solutions to the three systems are
Ch1_19
Homework
Exercise 1.1 :1, 2, 4, 5, 6, 7, 10, 13
Ch1_20
1-2 Gauss-Jordan EliminationDefinitionA matrix is in reduced echelon form ( 簡化梯式 ) if
1. Any rows consisting entirely of zeros are grouped at the bottom of the matrix.
2. The first nonzero element of each other row is 1. This element is called a leading 1.
3. The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row.
4. All other elements in a column that contains a leading 1 are zero.
Ch1_21
Examples for reduced echelon form
10000
04300
03021
9100
3010
7001
3100
0000
4021
000
210
801
() ()() ()
利用一連串的 elementary row operations ,可讓任何矩陣變成 reduced echelon form
The reduced echelon form of a matrix is unique.
隨堂作業: 2(b)(d)(h)
Ch1_22
Gauss-Jordan Elimination
System of linear equations augmented matrix reduced echelon from solution
Ch1_23
41200
22200
43111
4)R1(R3
610001110052011
R2)2(R3R2R1
Example 1Use the method of Gauss-Jordan elimination to find reduced echelon form of the following matrix.
1211244129333
22200
Solution
1211244
22200
129333
R2R1
pivot ( 樞軸,未來的 leading 1)
12112442220043111
R13
1
pivot
6100050100
170011
R3R22)R3(R1
The matrix is the reduced echelon form of the given matrix.
41200
11100
43111
R221
Ch1_24
Example 2Solve, if possible, the system of equations
7537429333
321
321
321
xxxxxxxxx
Solution (請先自行練習 )
715374123111
715374129333
R131
242012103111
3)R1(R3
2)R1(R2
000012104301
R2R3
R2R11243
1243
32
31
32
31
xxxx
xxxx
The general solution to the system is
.parameter) a (callednumber real is where,
12
43
3
2
1
rrx
rx
rx
隨堂作業: 5(c)
Ch1_25
Example 3Solve the system of equations
44694214232
58101242
4321
4321
4321
xxxxxxxxxxxx
Solution(自行練習 )
446942142321295621
44694214232158101242
R121
144300153300295621
2)R1(R3
R1R2
14430051100
295621
R231
110005110011021
3R2R3
6)R2(R1
110006010020021
R3R2
1)R3(R1
. somefor ,
1
6
22
1
6
22
4
3
2
1
4
3
21
r
x
x
rx
rx
x
x
xx
變數個數 >方程式個數 many sol.
Ch1_26
Example 4Solve the system of equations
432636242232
54321
54321
54321
xxxxxxxxxxxxxxx
Solution(自行練習 )
642000210000213121
431121636242213121
R1R3
2)R1(R2
210000
642000213121
R3R2
210000321000213121
2R21
210000
321000750121
3)R2(R1
210000101000300121
2)R3(R2
5R3R1
. and somefor ,2
,1 , ,32
21
32
5
432
1
5
4
321
srx
xsxrxsrx
xx
xxx
Ch1_27
Example 5This example illustrates a system that has no solution. Let us try to solve the system
12232
452232
32
321
321
321
xxxxxxxxxxx
Solution(自行練習 )
1220210063303211
R3R2
1220210021103211
R231
5400210021101101
2)R2(R4
R2R1
13000210040103001
4)R3R(R4
R3R2
1)R3(R1
1000210040103001
R4131
The system has no solution.
1220633021003211
1220312145223211
1)R1(R3
2)R1(R2
0x1+0x2+0x3=1隨堂作業: 5(d)
Ch1_28
Homogeneous System of linear Equations
Definition A system of linear equations is said to be homogeneous ( 齊
次 ) if all the constant terms ( 等號右邊的常數項 ) are zeros.
Example:
0632
052
321
321
xxx
xxx
Observe that is a solution. 0 ,0 ,0 321 xxx
Theorem 1.1
A system of homogeneous linear equations in n variables always has the solution x1 = 0, x2 = 0. …, xn = 0. This solution is called the trivial solution.
Ch1_29
Homogeneous System of linear Equations
Theorem 1.2
A system of homogeneous linear equations that has more variables than equations has many solutions.
Note. 除 trivial solution 外,可能還有其他解。
隨堂作業: 8(e)
0410
0301
0632
0521
0632
052
321
321
xxx
xxx
The system has other nontrivial solutions.
rxrxrx 321 ,4 ,3
Example:
Ch1_30
Homework
Exercise 1.2:2, 5, 8, 14
(Section 1.3 跳過 )