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Lecture 1
Systems of Linear Equations
Today- Course Overview- Introduction to Systems of Linear Equations- Matrices, Gaussian Elimination and Gauss-Jordan EliminationReading Assignment: Chapter 1 of TextHomework #1 Assigned (Due 10/05)
Elementary Linear AlgebraR. Larsen et al. (5 Edition) TKUEE 翁慶昌 -NTUEE SCC_09_2007
Lecture 1
Systems of Linear Equations
Next Time
- Operation with Matirces
- Properties of Matrix Operations
- Inverse of a Matrix
Reading Assignment: Secs 2.1-2.3 of Textbook
Elementary Linear AlgebraR. Larsen et al. (5 Edition) 翁慶昌 -SCC_09_2007
1 - 3
Purchasing Mix Problem
Mr. Wang has NT$1,000,000 and he wants spends all the money to buy 100 animals
Price tags:
NT$ 5000/sheep
NT$ 10,000/cow
NT$100,000/horse
Q: Formulate the problem mathematically?
1 - 4
Circuit Analysis
1 - 5
Examples from Your Daily Life ?
1 - 6
Algebra and Linearity
Algebra
Real Numbers or Sets
Binary Operations
Identity elements
Inverse elements
Associativity
Commutativity Linearity of a function
Additivity f(x+y) = f(x)+f(y)
Homogeneity f(ax) = af(x)
1 - 7
1.0.1 History of Linear Algebra
http://en.wikipedia.org/wiki/Linear_Algebra#History A Brief History of Linear Algebra and Matrix Theory.htm nla-1.ppt
1 - 8
1.0.2 Course Overview (Syllabus)
LA_4_mgmt_Syllabus_2007.doc
1 - 9
1.1 Introduction to Systems of Linear Equations
a linear equation in n variables:
a1,a2,a3,…,an, b: real number
a1: leading coefficient
x1: leading variable Notes:
(1) Linear equations have no products or roots of variables and
no variables involved in trigonometric, exponential, or
logarithmic functions.
(2) Variables appear only to the first power.
1 - 10
Ex 1: (Linear or Nonlinear)
723 )( yxa 2 21
)( zyxb
0102 )( 4321 xxxxc 221 4)
2sin( )( exxd
2 )( zxye 42 )( yef x
032sin )( 321 xxxg 411
)( yx
h
powerfirst not the
lExponentia
functions rictrigonomet
Linear
Linear Linear
Linear
Nonlinear
Nonlinear
Nonlinear
Nonlinear
powerfirst not the
1 - 11
a solution of a linear equation in n variables:
Solution set:
the set of all solutions of a linear equation
bxaxaxaxa nn 332211
,11 sx ,22 sx ,33 sx , nn sx
bsasasasa nn 332211s.t.
1 - 12
Ex 2 : (Parametric representation of a solution set)
42 21 xx
If you solve for x1 in terms of x2, you obtain
By letting you can represent the solution set as
And the solutions are or Rttt |) ,24(
,24 21 xx
tx 2
241 tx
Rsss |)2 ,( 21
a solution: (2, 1), i.e. 1,2 21 xx
1 - 13
a system of m linear equations in n variables:
mnmnmmm
n
nn
nn
bxaxaxaxa
bxaxaxaxabxaxaxaxabxaxaxaxa
332211
333333232131
22323222121
11313212111
Consistent:
A system of linear equations has at least one solution. Inconsistent:
A system of linear equations has no solution.
1 - 14
Notes:
Every system of linear equations has either
(1) exactly one solution,
(2) infinitely many solutions, or
(3) no solution.
1 - 15
Ex 4: (Solution of a system of linear equations)
(1)
(2)
(3)
13
yxyx
6223
yxyx
13
yxyx
solution oneexactly
number inifinite
solution no
lines ngintersecti two
lines coincident two
lines parallel two
1 - 16
Ex 5: (Using back substitution to solve a system in row echelon form)
(2)(1)
252
yyx
Sol: By substituting into (1), you obtain2y
15)2(2
xx
The system has exactly one solution: 2 ,1 yx
1 - 17
Ex 6: (Using back substitution to solve a system in row echelon form)
(3)(2)(1)
253932
zzyzyx
Sol: Substitute into (2) 2z
15)2(3
yy
and substitute and into (1)1y 2z
19)2(3)1(2
xx
The system has exactly one solution:2 ,1 ,1 zyx
1 - 18
Equivalent:
Two systems of linear equations are called equivalent
if they have precisely the same solution set.
Notes:
Each of the following operations on a system of linear
equations produces an equivalent system.
(1) Interchange two equations.
(2) Multiply an equation by a nonzero constant.
(3) Add a multiple of an equation to another equation.
1 - 19
Ex 7: Solve a system of linear equations (consistent system)
(3)(2)(1)
17552
43932
zyxyx
zyx
Sol:
(4) 17552
53932
(2)(2)(1)
zyxzyzyx
(5)
153932
(3)(3)2)((1)
zyzyzyx
1 - 20
So the solution is (only one solution)2 ,1 ,1 zyx
(6)
4253932
(5)(5)(4)
zzyzyx
253932
)6((6) 21
zzyzyx
1 - 21
Ex 8: Solve a system of linear equations (inconsistent system)
(3)(2)(1)
13222213
321
321
321
xxxxxxxxx
Sol:
)5()4(
24504513
(3)(3))1((1)
(2)(2)2)((1)
32
32
321
xxxxxxx
1 - 22
So the system has no solution (an inconsistent system).
2004513
)5()5()1()4(
32
321
xxxxx
statement) false (a
1 - 23
Ex 9: Solve a system of linear equations (infinitely many solutions)
(3)(2)(1)
13130
21
31
32
xxxxxx
Sol:
(3)(2)(1)
13013
)2()1(
21
32
31
xxxxxx
(4)
033013
(3)(3)(1)
32
32
31
xxxxxx
1 - 24
013
32
21
xxxx
then
,
,
,13
3
2
1
tx
Rttx
tx
tx 3let
,32 xx 31 31 xx
So this system has infinitely many solutions.
1 - 25
Keywords in Section 1.1:
linear equation: 線性方程式 system of linear equations: 線性方程式系統 leading coefficient: 領先係數 leading variable: 領先變數 solution: 解 solution set: 解集合 parametric representation: 參數化表示 consistent: 一致性 ( 有解 ) inconsistent: 非一致性 ( 無解、矛盾 ) equivalent: 等價
1 - 26
(4) For a square matrix, the entries a11, a22, …, ann are called
the main diagonal entries.
1.2 Gaussian Elimination and Gauss-Jordan Elimination
mn matrix:
mnmmm
n
n
n
aaaa
aaaaaaaaaaaa
321
3333231
2232221
1131211
rows m
columns n
(3) If , then the matrix is called square of order n.nm
Notes:
(1) Every entry aij in a matrix is a number.(2) A matrix with m rows and n columns is said to be of size mn
.
1 - 27
Ex 1: Matrix Size
]2[
0000
21
031
4722
e
11
22
41
23
Note:
One very common use of matrices is to represent a system
of linear equations.
1 - 28
a system of m equations in n variables:
mnmnmmm
nn
nn
nn
bxaxaxaxa
bxaxaxaxabxaxaxaxabxaxaxaxa
332211
33333232131
22323222121
11312212111
mnmmm
n
n
n
aaaa
aaaaaaaaaaaa
A
321
3333231
2232221
1131211
mb
bb
b2
1
nx
xx
x2
1
bAx Matrix form:
1 - 29
Augmented matrix:
][ 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
Coefficient matrix:
1 - 30
Elementary row operation:
jiij RRr :(1) Interchange two rows.
iik
i RRkr )(:)( (2) Multiply a row by a nonzero constant.
jjik
ij RRRkr )(:)((3) Add a multiple of a row to another row.
Row equivalent:
Two matrices are said to be row equivalent if one can be obtained
from the other by a finite sequence of elementary row operation.
1 - 31
Ex 2: (Elementary row operation)
143243103021
143230214310
12r
212503311321
212503312642 )(
121
r
8133012303421
251212303421 )2(
13r
1 - 32
Row-echelon form: (1, 2, 3)
(1) All row consisting entirely of zeros occur at the bottom
of the matrix.
(2) For each row that does not consist entirely of zeros,
the first nonzero entry is 1 (called a leading 1).
(3) For two successive nonzero rows, the leading 1 in the higher
row is farther to the left than the leading 1 in the lower row.
Reduced row-echelon form: (1, 2, 3, 4)
(4) Every column that contains a leading 1 has zeros everywhere
else.
1 - 33
form)echelon
-row (reduced
form)
echelon -(row
Ex 4: (Row-echelon form or reduced row-echelon form)
210030104121
10000410002310031251
000031005010
0000310020101001
310011204321
421000002121
form)
echelon -(rowform)echelon
-row (reduced
1 - 34
Gaussian elimination:
The procedure for reducing a matrix to a row-echelon form.
Gauss-Jordan elimination:
The procedure for reducing a matrix to a reduced row-echelon
form.
Notes:
(1) Every matrix has an unique reduced row echelon
form. (2) A row-echelon form of a given matrix is not unique.
(Different sequences of row operations can produce
different row-echelon forms.)
1 - 35
The first nonzerocolumn
Produce leading 1
Zeros elements below leading 1
leading 1
Produce leading 1
The first nonzero column
Ex: (Procedure of Gaussian elimination and Gauss-Jordan elimination)
1565422812610421270200
1565421270200281261042
12r
15654212702001463521
)(1
21
r
2917050012702001463521)2(
13r
Submatrix
1 - 36
Zeros elements below leading 1
Zeros elsewhere
leading 1
Produceleading 1
leading 1
1 2 5 3 6 14
0 0 1 0 7 2 6
0 0 5 0 17 29
)(2
21r
12100006270100
1463521)5(23
r
2100006270100
1463521)2(3r
210000100100703021)6(
31r )(
3227
r )5(21r
Submatrix
form)echelon -(rowform)echelon
-row (reduced
1 - 37
Ex 7: Solve a system by Gauss-Jordan elimination method (only one solution)
1755243932
zyxyx
zyx
Sol:matrix augmented
1755240319321
111053109321)2(
131
12 , rr
2100531093211
23r
210010101001)9(
31)3(
322
21 , , rrr
211
zy
x
form)echelon -(row
form)echelon -row (reduced
1 - 38
Ex 8: Solve a system by Gauss-Jordan elimination method (infinitely many solutions)
1 530242
21
311
xxxxx
13102501
)2(21
)1(2
)3(12
)(1 ,,,2
1 rrrr
is equations of system ingcorrespond the
13 25
32
31
xxxx
3
21
variablefree
, variableleading
x
xx
::
Sol:
10530242
matrix augmented
form)echelon
-row (reduced
1 - 39
32
31
3152
xxxx
Let tx 3
,
,31
,52
3
2
1
tx
Rttx
tx
So this system has infinitely many solutions.
1 - 40
Homogeneous systems of linear equations:
A system of linear equations is said to be homogeneous
if all the constant terms are zero.
0
0 0 0
332211
3333232131
2323222121
1313212111
nmnmmm
nn
nn
nn
xaxaxaxa
xaxaxaxaxaxaxaxaxaxaxaxa
1 - 41
Trivial solution:
Nontrivial solution:
other solutions
0321 nxxxx
Notes:
(1) Every homogeneous system of linear equations is consistent.
(2) If the homogenous system has fewer equations than variables,
then it must have an infinite number of solutions. (3) For a homogeneous system, exactly one of the following is true.
(a) The system has only the trivial solution.
(b) The system has infinitely many nontrivial solutions in
addition to the nontrivial solution.
1 - 42
Ex 9: Solve the following homogeneous system
020
321
321
xxxxxx
01100201
)1(21
)(2
)2(12 ,, 3
1
rrr
Let tx 3
Rttxtxtx , , ,2 321
solution) (trivial 0,0When 321 xxxt
Sol:
03120311
matrix augmented
form)echelon
-row (reduced
3
21
variablefree
, variableleading
x
xx
::
1 - 43
Keywords in Section 1.2:
matrix: 矩陣 row: 列 column: 行 entry: 元素 size: 大小 square matrix: 方陣 order: 階 main diagonal: 主對角線 augmented matrix: 增廣矩陣 coefficient matrix: 係數矩陣
1 - 44
elementary row operation: 基本列運算 row equivalent: 列等價 row-echelon form: 列梯形形式 reduced row-echelon form: 列簡梯形形式 leading 1: 領先 1 Gaussian elimination: 高斯消去法 Gauss-Jordan elimination: 高斯 - 喬登消去法 free variable: 自由變數 homogeneous system: 齊次系統 trivial solution: 顯然解 nontrivial solution: 非顯然解