chap. 1 systems of linear equations 1.1 introduction to systems of linear equations 1.2 gaussian...

Chap. 1Chap. 1Systems of Linear Systems of Linear

EquationsEquations

1.1 Introduction to Systems of Linear Equations

1.2 Gaussian Elimination and Gauss-Jordan Elimination

1.3 Applications of Systems of Linear Equations

Ming-Feng Yeh Chapter 1 1-2

Chapter ObjectivesChapter Objectives Recognize, graph, and solve a system of linear equations

in n variables. Use back substitution to solve a system of linear

equations. Determine whether a system of linear equations is

consistent or inconsistent. Determine if a matrix is in row-echelon form or reduced

row-echelon form. Use element row operations with back substitution to

solve a system in row-echelon form. Use elimination to rewrite a system in row echelon form.


Chapter Objective (cont.)Chapter Objective (cont.) Write an augmented or coefficient matrix from a system

of linear equation, or translate a matrix into a system of linear equations.

Solve a system of linear equations using Gaussian elimination with back-substitution.

Solve a homogeneous system of linear equations. Set up and solve a system of linear equations to fit a

polynomial function to a set of data points, as well as to represent a network.


Linear Equations in n VariablesLinear Equations in n Variables A linear equations in n variables x1, x2, …, xn has the form:

a1x1 + a2x2 + … + anxn = b Coefficients: a1, a2, …, an real number Constant term: b real number Leading Coefficient: a1

Leading Variable: x1

Linear equations have no products or roots of variables and no variables involved in trigonometric, exponential or logarithmic functions.

Variables appear only to the first power.

1.1 1.1 IntroductionIntroduction


Example 1Example 1 Linear Equations:

Nonlinear Equations:

Section 1-1

,723 yx

,221 zyx

,0102 4321 xxxx

.4)(sin 2212 exx

,2 zxy

,42 ye x

,032sin 321 xxx

.411 yx

True?

Product of variables

involved in exponential

involved in trigonometric

Not the first power


Example 2Example 2

Parametric Representation of a Solution Set Solve the linear equation x1 + 2x2 = 4

Sol: x1 = 4 2x2

Variable x2 is free (it can take on any real value).

Variable x1 is not free (its value depends on the value of x2).

By letting x2 = t (t: the third variable, parameter),

you can represent the solution set as

參數個數 = 變數個數方程式列數

Rttx

tx

,

,24

2

1

Section 1-1


Example 3Example 3

Parametric Representation of a Solution Set Solve the linear equation 3x + 2y z = 3

Sol: Choosing y and z to be the free variables

Letting y = s and z = t, you obtain the parametric representation

.1 31

32 zyx

Rtstzsy

tsx

,

.,

,1 31

32

Infinite number of solutions

Section 1-1


Systems of Linear EquationsSystems of Linear Equations A system of m linear equations in n variables is a set of

m equations, each of which is linear in the same n variables:

The double-subscript notation indicates that aij is the coefficient of xj in the ith equation.

A system of linear equations has exactly one solution, an infinite number of solutions, or no solution.

A system of linear equations is called consistent if it has at least one

solution and inconsistent if it has no solution.

mnmnmm

nn

nn

bxaxaxa

bxaxaxabxaxaxa

...

...

...

2211

22222121

11212111

Section 1-1


Example 4Example 4

Systems of two equations in two variables Solve the following systems of linear equations, and graph

each system as a pair of straight lines.

.2,11

3)(

yxyxyx

a

.,,36223

)( Rttytxyxyx

b

solution no 13

)(

yxyx

c

y

x

x

y

x

y

Two intersecting lines

Two coincident lines

Two parallel lines

Section 1-1


Solving a system of linear equations Solving a system of linear equations Row-echelon form: it follows a stair-step pattern and has

leading coefficient of 1. Using back-substitution to solve a system in row-echelon

form. Example 6.Example 6.

3 Eq.22 Eq.531 Eq.932

zzyzyx 1. From Eq. 3 you already know the value of z.

2. To solve for y, substitute z = 2 into Eq. 2 to obtain y = 1. 3. Substitute z = 2 and y = 1 into Eq. 1 to obtain x = 1.

Section 1-1


Equivalent SystemsEquivalent Systems Two systems of linear equations are called equivalent if

they have precisely the same solution set. 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.

Gaussian elimination: Rewriting a system of linear equations in row-echelon form usually involves a chain of equivalent systems, each of each is obtained by using one of the three basic operations.

Section 1-1


Example 7Example 7A system with exactly one solution

Solve the system17552

43

932

zyx

yx

zyx

17552

53

932

zyx

zy

zyx

1

53

932

zy

zy

zyx

42

53

932

z

zy

zyx

Adding the first equation to the second produces a new second equation.

Adding –2 times the first equation to the third equation produces a new third

equation.

(2)

Adding the second equation to the third equation produces a new third equation.

The solution is x = 1, y = 1, and z = 2.

(1/2)

Section 1-1


Example 8Example 8An Inconsistent System Solve the system

132

222

13

321

321

321

xxx

xxx

xxx

132

045

13

321

32

321

xxx

xx

xxx

245

045

13

32

32

321

xx

xx

xxx

20

045

13

32

321

xx

xxx

Adding –2 times the first equation to the second produces a new second equation.

(2)

(1)

Adding –1 times the first equation to the third produces a new third equation.

(1)

Adding –1 times the 2nd equation to the 3rd produces a new 3rd equation.

Because the third “equation” is a false statement, this system has no solution. Moreover, because this system is equivalent to the original system, you can conclude that the original system also has no system.

Section 1-1


Example 9Example 9A system with an infinite number of solutions

Solve the system

1313

0

21

31

32

xxxxxx

130

13

21

32

31

xxxxxx

000

13

32

31

xxxx

0330

13

32

32

31

xxxxxx

The first two equations are interchanged.

Adding the first equation to the third produces a new third equation.

Adding –3 times the 2nd equation to the 3rd produces a new 3rd equation.

(3)unnecessary

Let x3 = t, t R

txx

txx

32

31 3131

Section 1-1


1.2 Gaussain Elimination and1.2 Gaussain Elimination and Gauss-Jordan Elimination Gauss-Jordan Elimination

Definition: Matrix

If m and n are positive integers,

then an mn matrix is a rectangular array

in which each entry, aij, of the matrix is a number. An mn matrix has m rows (horizontal lines) and n columns (vertical lines). The entry aij is located in the ith row and the jth column. A matrix with m rows and n columns (an mn matrix) is said to be of size mn. If m = n, the matrix is called square of order n. For a square matrix, the entries a11, a22, a33, … are called the main diagonal

entries.

mnmmm

n

n

n

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211


Augmented/Coefficient MatrixAugmented/Coefficient Matrix The matrix derived from the coefficients and constant

terms of a system of linear equations is called the augmented matrix of the system.

The matrix containing only the coefficients of the system is called the coefficient matrix of the system.

System Augmented Matrix Coefficient Matrix

642

33

534

zx

zyx

zyx

6402

3131

5341

402

131

341

Section 1-2

x y z const.


Elementary Row Operations Elementary Row Operations Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row.

Two matrices are said to be row-equivalent

if one can be obtained from the other by a finite sequence of elementary row operations.

Section 1-2


Example 3Example 3 Using Elementary Row Operation to Solve a System

Linear System Associated Augmented matrix

17552

43

932

zyx

yx

zyx

17552

4031

9321 R2+R1R2

17552

5310

9321

1110

5310

9321

4200

5310

9321

2100

5310

9321

R3+(2)R1R3

(2) R3+R2R3

0.5

0.5R3R32z

153 yzy

1932 xzyx

Section 1-2


Row-Echelon Form of a MatrixRow-Echelon Form of a MatrixA matrix in row-echelon form has the following properties. All rows consisting entirely of zeros occur at the bottom

of the matrix. For each row that does not consist entirely of zeros, the

first nonzero entry is 1 (called a leading 1). For two successive (nonzero) rows, the leading 1 in the

higher row is father to the left than the leading 1 in the lower row.

Remark: A matrix in row-echelon form is in reduced row-echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1.

Section 1-2


Example 4Example 4 In row-echelon form

Not in row-echelon form

0000

3100

2010

1001

10000

41000

23100

31251

0000

3100

5010

4210

0000

2121

3100

1120

4321

Section 1-2


Gaussian Elimination withGaussian Elimination withBack-Substitution Back-Substitution

Write the augmented matrix of the system of linear equations.

Use elementary row operations to rewrite the augmented matrix in row-echelon form.

Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution.

Section 1-2


Example 5Example 5A system with exactly one solution

Solve the system1974

2342

22

32

4321

4321

321

432

xxxx

xxxx

xxx

xxx

211660

63300

32110

20121

191741

23142

32110

20121(2)

191741

23142

20121

32110

6

3913000

63300

32110

20121

)31()131(

34 x

12 343 xxx

232 2432 xxxx

122 1321 xxxx

Section 1-2


Example 6Example 6A system with no solution

Solve the system

123

4532

6

42

321

321

31

321

xxx

xxx

xx

xxx

1123

4512

6101

4211

11750

2000

2110

4211

11750

4110

2110

4211(1)

(2)(3)

0 = 2 … ???The original system of linear equationsis inconsistent.

Section 1-2


Gauss-Jordan Elimination Gauss-Jordan Elimination Continues the reduction process until a reduced row-

echelon form is obtained. Example 7:

Use Gauss-Jordan elimination to solve the system

17552

43

932

zyx

yx

zyx

2100

5310

9321In Ex. 3 2

2100

5310

19901

(3)

(9)

2100

1010

1001

2

1

1

z

y

x

Section 1-2


Example 8Example 8 A System with an Infinite Number of Solutions

Solve the system of linear equations153

0242

21

321

xx

xxx

1310

2501

1310

0121

1053

0121

1053

0242 )21( )3(

)1(

Let x3 = t, t R

txx

txx

3131

5252

32

31

Section 1-2


Homogeneous Systems of Linear Homogeneous Systems of Linear Equations Equations

Each of the constant terms is zero.

A homogeneous system must have at least one solution. Trivial (obvious) solution: all variables in a homogeneous

system have the value zero, then each of the equation must be satisfied.

0

0

0

0

332211

3333232131

2323222121

1313212111

nmnmmm

nn

nn

nn

xaxaxaxa

xaxaxaxa

xaxaxaxa

xaxaxaxa

Section 1-2


Example 9Example 9 Solve the system of linear equations

032

03

321

321

xxx

xxx

0110

0201

0110

0311

0330

0311

0312

0311

)31(

)1(

)1(

Let x3 = t, t R

txx

txx

32

31 22

Section 1-2


Theorem 1.1Theorem 1.1

The Number of Solutions of a Homogeneous System Every homogeneous system of linear equations is

consistent. Moreover, if the system has fewer equations than variables, then it must have an infinite number of solutions.

Section 1-2


1.3 Applications of Systems of1.3 Applications of Systems of Linear Equations Linear Equations Polynomial Curve Fitting

fit a polynomial function toa set of data points in the plane. n points: Polynomial function:

Network Analysisfocus on networks and Kirchhof’s Laws for electricity.

),(...,),,(),,( 2211 nn yxyxyx

11

2210)(

nn xaxaxaaxp


Network AnalysisNetwork Analysis Networks composed of branches and junctions are used as

models in the fields as diverse as economics, traffic analysis, and electrical engineering.

The total flow into a junction is equal to the total flow out of the junction.

Example.

Section 1-3

251x

2x

2521 xx


Example 5Example 5

45

51

32

43

21

10

10

20

20

20

xx

xx

xx

xx

xx

1011000

1010001

2000110

2001100

2000011

000000

1011000

1010100

3010010

1010001

10

10

30

10

,

4

3

2

1

5

tx

tx

tx

tx

Rttx

Section 1-3


Kirchhoff’s Laws Kirchhoff’s Laws All the current flowing into a junction must flow

out of it. (KCL) The sum of the products IR (I is the current and R

is the resistance) around a closed path is equal to the total voltage in the path. (KVL)

A closed path is a sequence of branches such that the beginning point of the first branch coincides with the end point of the last branch.

Section 1-3


Example 6Example 6

or : 231 III

Path 1: 723 212211 IIIRIR

Path 2: 842 323322 IIIRIR

1100

2010

1001

8420

7023

0111

amp1

amp2

amp1

3

2

1

I

I

I

Section 1-3


Example 7Example 7

See pp. 37-38

in the textbook

Section 1-3

chap. 1 systems of linear equations 1.1 introduction to systems of linear equations 1.2 gaussian...

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