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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
Elimination
Reading Assignment: Chapter 1 of Text
Homework #1 Assigned (Due 10/05)
Elementary Linear Algebra
R. 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 Algebra
R. Larsen et al. (5 Edition)
翁慶昌-SCC_09_2007
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?
1-3
Circuit Analysis
1-4
Examples from Your Daily Life ?
1-5
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
1-6
1.0.2 Course Overview (Syllabus)

LA_4_mgmt_Syllabus_2007.doc
1-7
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.
1-8

Ex 1: (Linear or Nonlinear)
1
( b) x  y   z  2
2
Linear (a ) 3x  2 y  7
Linear

2
(
c
)
x

2
x

10
x

x

0
(
d
)
(
sin
)
x

4
x

e
1
2
3
4
Linear
Linear
1
2
2
Exponentia l
x
(
f
)
e
 2y  4
(
e
)
xy

z

2
Nonlinear
Nonlinear
not the first power
Nonlinear ( g ) sinx1  2 x2  3x3  0
trigonomet ric functions
1-9
1 1
(h)   4
x y
Nonlinear
not the first power

a solution of a linear equation in n variables:
a1 x1  a2 x2  a3 x3    an xn  b
x1  s1 , x2  s2 , x3  s3 , , xn  sn
s.t.

a1s1  a2 s2  a3 s3    an sn  b
Solution set:
the set of all solutions of a linear equation
1 - 10

Ex 2： (Parametric representation of a solution set)
x1  2 x2  4
a solution: (2, 1), i.e. x1  2, x2  1
If you solve for x1 in terms of x2, you obtain
x1  4  2 x2 ,
By letting x2  t you can represent the solution set as
x1  4  2t
And the solutions are (4  2t, t ) | t  R or ( s, 2  12 s) | s  R
1 - 11

a system of m linear equations in n variables:
a11 x1  a12 x2  a13 x3
a21 x1  a22 x2  a23 x3
a31 x1  a32 x2  a33 x3

am1 x1  am 2 x2  am 3 x3

   a1n xn  b1
   a2 n xn  b2
   a3n x3  b3
   amn xn  bm
Consistent:
A system of linear equations has at least one solution.

Inconsistent:
A system of linear equations has no solution.
1 - 12

Notes:
Every system of linear equations has either
(1) exactly one solution,
(2) infinitely many solutions, or
(3) no solution.
1 - 13

Ex 4: (Solution of a system of linear equations)
(1)
(2)
(3)
x  y  3
x  y  1
two intersecti ng lines
exactly one solution
x  y  3
2x  2 y  6
two coincident lines
inifinite number
x  y  3
x  y  1
two parallel lines
no solution
1 - 14

Ex 5: (Using back substitution to solve a system in row echelon form)
x  2y  5
y  2
(1)
(2)
Sol: By substituting y  2 into (1), you obtain
x  2( 2)  5
x  1
The system has exactly one solution: x  1, y  2
1 - 15

Ex 6: (Using back substitution to solve a system in row echelon form)
x  2 y  3z  9
y  3z  5
z  2
(1)
(2)
(3)
Sol: Substitute z  2 into (2)
y  3(2)  5
y  1
and substitute y  1 and z  2 into (1)
x  2( 1)  3(2)  9
x  1
The system has exactly one solution:
x  1, y  1, z  2
1 - 16

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 - 17

Ex 7: Solve a system of linear equations (consistent system)
x  2 y  3z  9
 x  3y
 4
2 x  5 y  5z  17
Sol:
(1)
(2)
(3)
(1)  (2)  (2)
x  2 y  3z  9
y  3z  5
2 x  5 y  5z  17
(4)
(1)  ( 2)  (3)  (3)
x  2 y  3z  9
y  3z  5
 y  z  1
1 - 18
(5)
(4)  (5)  (5)
x  2 y  3z  9
y  3z  5
2z  4
(6)
(6)  12  (6)
x  2 y  3z  9
y  3z  5
z  2
So the solution is x  1, y  1, z  2 (only one solution)
1 - 19

Ex 8: Solve a system of linear equations (inconsistent system)
x1  3x2  x3  1
2 x1  x2  2 x3  2
x1  2 x2  3x3   1
(1)
(2)
(3)
Sol: (1)  ( 2)  (2)  (2)
(1)  ( 1)  (3)  (3)
x1  3x2  x3  1
5 x 2  4 x3  0
5 x 2  4 x3   2
1 - 20
( 4)
(5)
( 4)  ( 1)  (5)  (5)
x1  3x2  x3  1
5 x 2  4 x3  0
0  2
(a false statement)
So the system has no solution (an inconsistent system).
1 - 21

Ex 9: Solve a system of linear equations (infinitely many solutions)
x1
 x1
x 2  x3  0
 3 x3   1
 3 x2
 1
Sol: (1)  ( 2)
x1
(1)
(2)
(3)
 3 x3   1
 x3  0
 1
(1)
(2)
(3)
(1)  (3)  (3)
x1
 3 x3   1
x 2  x3  0
3 x 2  3 x3  0
(4)
 x1
x2
 3 x2
1 - 22
x1
x2
 3 x2
 x3


1
0
 x2  x3 , x1  1  3x3
let x3  t
then x1  3t  1,
x2  t ,
tR
x3  t ,
So this system has infinitely many solutions.
1 - 23
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 - 24
1.2 Gaussian Elimination and Gauss-Jordan Elimination

mn matrix:
 a11
 a21
 a31


am1

a13  a1n 
a23  a2 n 
a33  a3n  m rows


am 3  amn 
n columns
a12
a22
a32

am 2
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 mn .
(3) If m  n , then the matrix is called square of order n.
(4) For a square matrix, the entries a11, a22, …, ann are called
the main diagonal entries.
1 - 25

Ex 1:
Matrix
[ 2]
Size
1 1
0 0
0 0
22
1  3 0 1 

2 
 e  
2
2
  7 4 

1 4
3 2
Note:
One very common use of matrices is to represent a system
of linear equations.
1 - 26

a system of m equations in n variables:
a11 x1  a12 x2  a12 x3
a21 x1  a22 x2  a23 x3
a31 x1  a32 x2  a33 x3

am1 x1  am 2 x2  am 3 x3
Matrix form:
 a11
 a21
A   a31


am1
a12
a22
a32

am 2
   a1n xn  b1
   a2 n xn  b2
   a3n xn  b3
   amn xn  bm
Ax  b
a13  a1n 
a23  a2 n 
a33  a3n 


am 3  amn 
1 - 27
 x1 
 
x   x2 

 xn 
 b1 
 
b   b2 

bm 

Augmented matrix:
 a11
 a21
 a31


am1

a12
a22
a32

am 2
a13  a1n
a23  a2 n
a33  a3n
am 3  amn
b1 
b2 
b3   [ A b]


bm 
Coefficient matrix:
 a11
 a21
 a31


am1
a12
a22
a32

am 2
a13  a1n 
a23  a2 n 
a33  a3n   A


am 3  amn 
1 - 28

Elementary row operation:
(1) Interchange two rows.
rij : Ri  R j
(2) Multiply a row by a nonzero constant.
ri( k ) : (k ) Ri  Ri
(3) Add a multiple of a row to another row. r ( k ) : (k ) R  R  R
ij
i
j
j

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 - 29

Ex 2: (Elementary row operation)
 0 1 3 4
 1 2 0 3
 2  3 4 1
 2  4 6  2
 1 3  3 0
 5  2 1 2
 1 2  4 3
0 3  2  1
2 1 5  2
r12
( 12 )
1
r
r13( 2 )
 1 2 0 3
 0 1 3 4
 2  3 4 1
1  2 3  1
1 3  3 0
5  2 1 2
 1 2  4 3
0 3  2  1
0  3 13  8
1 - 30

Row-echelon form: (1, 2, 3)

Reduced row-echelon form: (1, 2, 3, 4)
(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.
(4) Every column that contains a leading 1 has zeros everywhere
else.
1 - 31

Ex 4: (Row-echelon form or reduced row-echelon form)
 1 2  1 4 (row - echelon
0 1 0 3
0 0 1  2 form)
0 1 0 5
0 0 1 3
0 0 0 0
 1  5 2  1 3
0 0 1 3  2 (row - echelon
0 0 0 1 4 form)
0 0 0 0 1
1
0
0
0
 1 2  3 4
0 2 1  1
0 0 1  3
 1 2  1 2
0 0 0 0
0 1 2  4
1 - 32
0
1
0
0
(reduced row echelon form)
0  1
0 2 (reduced row 1 3 echelon form)
0 0

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 - 33
Ex: (Procedure of Gaussian elimination and Gauss-Jordan elimination)

Produce leading 1
0 0  2 0 7 12
2 4  10 6 12 28
2 4  5 6  5  1
r12
2 4  10 6 12 28
0 0  2 0 7 12
2 4  5 6  5  1
The first nonzero column
( 12 )
1
r
leading 1
 1 2  5 3 6 14 r13( 2 )
0 0  2 0 7 12
2 4  5 6  5  1
Zeros elements below leading 1
1 - 34
Produce leading 1
6 14
1 2  5 3
7 12
0 0  2 0
0 0 5 0  17  29
The first nonzero Submatrix
column
leading 1
(  12 )
2
r
1 2  5 3
6 14 r23( 5)
0 0 1 0  7 2
6
0 0 5 0  17  29
1 2  5 3
6 14
0 0 1 0  7 2 6
0 0 0 0 1 2 1
Submatrix
Zeros elements below leading 1
Produce
leading 1
Zeros elsewhere
6 14 r ( 6 ) r ( 72 ) r ( 5)
r3( 2 )  1 2  5 3
0 0 1 0  7 2 6 31 32 21
0 0 0 0
1 2
 1 2 0 3 0 7
0 0 1 0 0 1
0 0 0 0 1 2
leading 1
(reduced row echelon form)
(row - echelon form)
1 - 35

Ex 7: Solve a system by Gauss-Jordan elimination method
(only one solution)
x  2 y  3z  9
 x  3y
 4
2 x  5 y  5z  17
Sol:
augmented matrix
 1  2 3 9 r121 , r13( 2 )
  1 3 0  4
 2  5 5 17
 1  2 3 9 r231
0 1 3 5
0  1 1  1
 1  2 3 9
0 1 3 5
0 0 1 2
(row - echelon form)
r212 , r32( 3) , r31( 9 )  1 0 0 1
0 1 0  1
0 0 1 2
x
(reduced row - echelon form)
1 - 36
 1
y
 1
z  2

Ex 8：Solve a system by Gauss-Jordan elimination method
(infinitely many solutions)
2 x1  4 x1  2 x3  0
3x1  5 x2
 1
Sol: augmented matrix
( 3)
( 1)
( 2 )
r
,
r
,
r
,
r
12
2
21
2 4  2 0
 1 0 5 2
 3 5 0 1
0 1  3  1
( 12 )
1
the correspond ing system of equations is
x1
 5 x3  2
x2
 3 x3   1
leading variable ：x1 , x2
free variable ： x3
1 - 37
(reduced row echelon form)
x1  2  5 x3
x 2   1  3 x3
Let x3  t
x1  2  5t ,
x2  1  3t ,
tR
x3  t ,
So this system has infinitely many solutions.
1 - 38

Homogeneous systems of linear equations:
A system of linear equations is said to be homogeneous
if all the constant terms are zero.
a11 x1  a12 x2 
a21 x1  a22 x2 
a31 x1  a32 x2 

am1 x1  am 2 x2 
a13 x3    a1n xn  0
a23 x3    a2 n xn  0
a33 x3    a3n xn  0
am 3 x3    amn xn  0
1 - 39

Trivial solution:
x1  x2  x3    xn  0


Nontrivial solution:
other solutions
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 - 40

Ex 9: Solve the following homogeneous system
x1  x2  x3  0
2 x1  x2  x3  0
Sol:
augmented matrix
(2)
12
( 13 )
2
r , r , r21(1)
 1  1 3 0
2 1 3 0
leading variable ：x1 , x2
free variable ： x3
 1 0 2 0 (reduced row 0 1  1 0 echelon form)
Let x3  t
x1  2t , x2  t , x3  t , t  R
When t  0, x1  x2  x3  0 (trivial solution)
1 - 41
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 - 42

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
1 - 43