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SYSTEMS OF LINEAR

EQUATIONS

Topic-1

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Real-Life Examples

Example 1: Very simple case

I am thinking of a number. If I add 3 to that number, I will get 7.What is the number?

Let x be the number in my mind

Adding 3 to x gives 7 The equation is 3 + x = 7

Example 2:

Taxi drivers usually charge a an initial fixed fee as part of usingtheir services. Then, for each mileage, they charge a certain

amount

Say for instance, the initial fee is 10 AED and each mileage cost 2AED

How will be model the total cost as an equation?

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Real-Life Examples (cont.)

Example 2 (cont.)

Solution

Let y be the total cost

Let N be the number of miles

So, the total cost = initial cost + number of miles x chargesper mile

That is, y = 4 + N x 2

So what can we infer?

These equations model the relationship betweentwo variables and the effect that a change on onevariable has on the other

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Why is this important?

Think of the following situations

the relationship between the price of a product and the quantityconsumers are willing to buy demand planning

how much of their products to sell at a price that maximizes profits supply curve

how does the change in one currency value affect the otherforeign exchange

In the more complex case of a manufacturers profit depends on:

Material cost, labor costs, transportation, overhead etc.

A realistic modeling of these relationships would involve all thesevariables

Mathematically, profit now is a function of several variables

http://www.ehow.com/facts_6027891_examples-equations-used-real-life.html

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Linear Algebra

In linear algebra we study the simplest functions of several variables, the onesthat are linear

Linear equations:

In n unknowns, linear equation is an equation that is of the form:

Which is the unknown?

x1, x

2, x

n

Then, what are a1

, a2

, an

and b?

Known, a1, a

2, a

n coefficients and b constant

The solution of a linear equation is any sequence s1, s2, sn of number suchthat the substitution of x

1= s

1, x

2= s

2, x

n= s

nsatisfies the equation

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Why linear?

Linear because:

the unknowns only appear to the first power

there arent any unknowns in the denominator

of a fraction. there are no products and/or quotients of

unknowns.

In the other way:

Unknowns only occur in numerators, they areonly to the first power and there are no productsor quotients of unknowns.

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Linear System

Systems of linear equations

collection of two or more linear equationsinvolving the same unknowns

Consider,

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Linear System

Unknowns: x1, x

2, x

n; coefficients: a

11, a

12,

Constants: b1, b

2, b

m

Here, the solution set to a system with nunknowns is a set of numbers so that thesubstitution of x

1= t

1, x

2= t

2, x

n= t

n

satisfies all the equation

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Graphical Interpretation

Let us consider:

Subject to

Coefficients of each equation not both zeros

Three possible solutions:

222121

1212111

bxaxa

bxaxa

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Geometrical Interpretation

Now lets see:

Represents planes

Possible solutions include:

The two planes might be coincident or they canintersect in a line;

Infinite solutions

The two planes can be parallel No solution

How about the same case for (3 x 3) system?

323222121

1313212111

xaxaxa

xaxaxa

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Geometric Interpretation

In general an (m x n) system of linearequations may have (a) infinitely manysolutions, (b) no solution or (c) a uniquesolution

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Classification of Linear Systems

There is a unique solution, in which case we say that thesystem is consistent and nonsingular. For a nonsingularsystem we must have that m = n, although the converse isnot true.

There are no solutions, in which case we say that thesystem is singularand inconsistent. Typically (but notalways!) an inconsistent system is one which is over-determined that is, where m > n.

There are infinitely many solutions, in which case we say

that the system is consistent but singular. Typically (but notalways!) a consistent, singular system is one which isunder-determined that is, where m < n.

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Differentiation

Consistent system has atleast one solution

Inconsistent system has no solution

A consistent system has either one solution or an infinitenumber of solutions (i.e. it is not possible for a linear

system to have, for example, exactly five solutions)

Homogeneous all constant terms are zero; otherwise non-homogeneous

Just-determined system: same equations and unknowns

Over-determined system: more equations than unknowns

Under-determined system: fewer equations than unknowns

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Matrices an introduction

Matrices provide a natural symbolic language to describelinear systems

Appropriate, convenient and powerful framework foranalyzing and solving linear problems

In general, an (m x n) matrix is a rectangular array of theform:

m rows; n columns

Have we seen the above notation somewhere?

mnmm

n

n

aaa

aaa

aaa

A

...

....

...

...

21

22221

11211

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Linear System - Matrixrepresentation

Let re-think the general equation of a linear system.

How about?

B is called the augmented matrix for the system denoted as[A | b]

mmnmm

n

n

baaa

baaa

baaa

B

...

.....

.....

...

...

21

222221

111211

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Linear System MatrixRepresentation

Where, A is called the coefficient matrix ofthe form,

And b is,

mnmm

n

n

aaa

aaa

aaa

A

...

....

...

...

21

22221

11211

mb

b

b

b.

2

1

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Example

Let us consider,

Here,

23

12

22

321

321

321

xxx

xxx

xxx

213

112

121

A

5

1

2

b

2131112

2121

]|[ bAB

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Solving Linear System ofEquations

Two steps involved in solving an (m x n) system ofequations are:

Reduction of the system (that is, elimination of variables)

Goal of reduction is to yield equivalent system of equations

Description of the set of solutions

What are we trying to achieve?

Obtain Equivalence

Two systems of linear equations in n unknowns areequivalent provided they have the same set of solutions

How to perform reduction? Elementaryoperations...

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Elementary Operations

High school method of solving linear equations

Possible elementary operations

Interchange two equations

Multiply an equation by non-zero scalarAdd a constant multiple of one equation to another

That is,

EiEj interchange i and j equations

kEi multiply ith equation by non zero scalar k

Ei+kEj k times the jth equation added to the ith equation

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Example

Let us consider,

Now the elementary operation E2 + E1 willproduce,

Now (1/3)E2 will give us,

Finally, E1 E2 will produce,

2

5

21

21

xx

xx

93

5

2

21

x

xx

3

5

2

21

x

xx

3

2

2

1

x

x

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Row Operations

But we are in a university, so lets think matrices

Elementary operations on rows of the matrix?

So, elementary row operations are:

Interchange 2 rows

Multiply a row by a non-zero scalar

Add a constant multiple of one row to another

That is,RiRj interchange i and j rows

kRi multiply ith row by non zero scalar k

Ri+kRj k times the jth row added to the ith row

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Solving a System of LinearEquations

Thus we can solve a linear system with thefollowing steps

Form the augmented matrix B for the system

Use elementary row operations to transform B to rowequivalent matrix Cthis represents a simpler system

Solve the simpler system represented by C

Gauss-Jordan Elimination method reductionprocess

Givensystem ofequations

Build theAugmented

Matrix

ReducedMatrix

ReducedSystem ofEquations

Solution

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Note:

Why use matrix?

Because we do not need the list of variables, the sequence ofsteps in the matrix method is easier to perform and record.

The objective of the Gauss-Jordan reductionprocess is to obtain a system of equations to apoint where we can immediately describe thesolution.

How do we know when the system has beensimplified as much as it can be?

The system has been simplified as much as possible when itis in the row echelon form

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Echelon Form

An (m x n) matrix B is in echelon form if:

All rows that consist entirely of zeros are grouped together atthe bottom of the matrix

In every non-zero row, the first entry (counting left to right) is

a 1 (call this a leading 1)

If the (i+1)st row contains non-zero entries, then the first non-zero entry is in a column to the right of the first non-zero entryin the ith row.

A matrix that is in echelon form is in reducedechelon form provided that the first non-zero entryin any row is the only non-zero entry in its column

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Examples

Echelon form

Reduced Echelon form

0000000

3100000

2121000

2348100

0203411

A

10000

56100

34110

B

3100

1010

2001

A

00000

43100

11021

B

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Reduction to Echelon Form

Conversion process to the Reduced Echelon form for an (mx n) matrix (cont.):

Locate the first (left most) column that contains a nonzero entry

If necessary, interchange the first row with another so that the firstnonzero column has a nonzero entry in the first row.

If a denotes the leading nonzero entry in row one, multiply each entry inrow one by (1/a). Thus leading nonzero entry in row one is 1)

Add appropriate multiples of row one to each of the remaining rows sothat every entry below the leading 1 in row one is zero.

Once done, ignore row one temporarily, and repeat above process ofthe submatrix this will produce the echelon form

To get to the reduced echelon form, proceed upward, add multiples ofeach nonzero row to the rows above in order to zero all entries abovethe leading 1

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Example

1101000

4820000

91131000

11831410

1000000

0410000

0101000

0300410

211761820

3324931230

91131000

4820000

211761820

4820000

91131000

3324931230

211761820

4820000

91131000

11831410

81230000

4820000

91131000

2300410

81230000

2410000

91131000

2300410

2000000

2410000

3101000

2300410

1000000

2410000

3101000

2300410

R1R3

R1(1/3)R1

R4->R4+2R1

R1->R1+R2R4->R4+R2

R3->(1/2)R3

R2->R2+(-3)R3R4->R4+(-3)R3

R4->(1/2)R4

R1->R1+2R4R2->R2+(-3)R4R3->R3+(-2)R4

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Reduction to Echelon Form

Let B be an (m x n) matrix. There is a unique (m x n) matrixsuch that:

C is in reduced echelon form

C is row equivalent to B

Solving System of Linear Equations

Create the augmented matrix of the system

Transform the augmented matrix to reduced echelon

Decode the reduced matrix to obtain equivalent system ofequations

By examining the reduced system, describe the solution ofthe original system

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Solve

612881063

1063

13522

28114342

54321

543

54321

54321

xxxxx

xxx

xxxxx

xxxxx

Recognizing an Inconsistent

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Recognizing an InconsistentSystem

Let [A|b] be the augmented matrix for an (m x n)linear system of equations.

If [A|b] is in reduced echelon form and if the lastnonzero row of [A|b] has its leading 1 in the last

column, then system of equations has no solution.

i.e.

last nonzero row of [A|b] has the form:

[0 0 0 0 1]

The equation for the row is 0x1 + 0x2+ + 0xn = 1

Consistent Systems of Linear

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Consistent Systems of LinearEquations

Goal is to deduce as much information as possible about the solutionset without actually solving the system.

Let [A|b] be the augmented matrix representing a (m x n) linear system

[A|b] can be simplified to the reduced row echelon form to a rowequivalent matrix [C|d]

[C|d], we can say:

Is inconsistent if and only if [C|d] has a row of the form [0 0 0 0 0 1]

Every variable corresponding to a leading 1 in [C|d] is a dependent variable (i.e.leading 1 variables can be expressed in terms of the independent or nonleading 1variables

If r denote the number of nonzero rows in [C|d],

then r

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Homogeneous Systems

Homogeneous system has

the form:

Such a linear system is

always consistent.i.e. x1=x2=x3=xn=0 is a solution

This solution is called the trivial or zero solution

Any other solution to such a homogeneous system is called

nontrivial solution

A homogeneous system can have a unique trivial solutionor also has non-trivial (infinitely many) solutions.

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Example

What are the possibilities for the solution setof

02663

0342

032

4321

4321

4321

xxxx

xxxx

xxxx

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Solution

02663

01342

03121

07300

05100

03121

08000

05100

08021

01000

00100

00021

0

0

02

4

3

21

x

x

xx

0

0

2

4

3

21

x

x

xx

R2-2R1, R3-3R1 R3-3R2,R1-R2 (1/8)R3,R1-8R3,R2+5R3