System of Linear Equation

7/21/2019 System of Linear Equation

http://slidepdf.com/reader/full/system-of-linear-equation-56da3def16742 1/91



Topic Outline1.Systems of Linear Equations

2. Matrices

3. Applications of systems of LinearEquations

Linear Equations



Definition 1: A linear equation in two variables x and y has

the form a1 x+a2 y= where ab, are

constants a1 , a1 and a2 and2b are not both zero.

Definition 2: A linear equation in three variables x , y, and z

has the form

a1 x+a2 y+a3 z =b where

are constants.a1 . a2 , a3 andb

Definition 3: A linear equation in n variables x andy has the form



a1 x1 +a2 x2 +a3 x 3 +...+an xn =b

where the coefficients a1, a2, a3 ,...,anandb

are real numbers. The number is the leading coefficient a1

and is the leading variable. x1

Systems of Linear EquationsDefinition 4: A system of m linear equations in n

variables x1, x2, …, xn is a set of m linearequations in the same n variables.



m n mn m m

n n

n n

b a a a

b a a a

b a a a

= + +

= + +

= + +

...

...

...

...

...

2 2 1 1

2 2 2 22 1 21

1 1 2 12 1 11



Types of Systems

Definition 5: A system of linear equations that

has a solution is called a consistent system

while a system of linear equations that has no

solution is called an inconsistent system.

Definition 6: An underdetermined system oflinear equations is one with more variables

than equations.

Example: x1 +2 x2 −3 x3 =4

2 x1 − x2 +4 x3 =−3



Definition 7: An overdetermined system of

linear equations is one with more equations

than variables.

x1 +3 x2 =5

Example: 2 x1 −2 x2 =−3

− x1 +7 x2 =0

Elementary Linear Algebra by Ron Larson and David Falvo

Systems of Equations in Two Unknowns

Independent system Inconsistent system Dependent system

One solution

No solution

Infinite solution



Solving Systems of LinearEquations in Two

Unknowns

1. Graphical

2. Substitution



3. Addition or Elimination

4. Cramer’s Rule

5. Gauss-Jordan Elimination

6. Use of Inverse of a Matrix

Solving Systems of Linear Equations

1.Graphical Method

Solve the system:




2.Substitution Method

y = 3 – 2

y = – – 6

Solution:

(-1,-5)




Solve the

system

+

y

=

10

2 + y = 15

Solution:



3.Addition Method

Solve thesystem

5x +3y = -19

8 x +3y =-25

Solution:



Introduction : Cramer’s Rule

Matrices

Definition 8: If m and n are positive integers, then an mxn matrix is a

rectangular array of numbers where aij is the entry located in the ith

row and jth column



Operations on Matrices

Addition of Matrices

⎥

⎥

⎥

⎥ ⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢ ⎢

⎢

⎣

⎡

mn 3 m 2 m 1 m

.

n 3 33 33 31

n 2 23 22 21

n 1 13 12 11

a ... a a a

. . . .

a ... a a a

a ... a a a

a ... a a a

m rows

n columns



Let A = [aij ] and B =[bij ] be of the same size.

Then A + B is the matrix obtained by adding

corresponding elements of A and B; that is

A + B = [aij ] + [bij ] = [aij + bij ]:

Example: ⎡⎣⎢14 52⎤⎦+⎡⎢⎣13 −41⎤⎥⎦=⎡⎢⎣14++31

52+−14⎥⎦⎤=⎡⎢⎣54 19⎤⎥⎦

⎥



Scalar Multiple of a Matrix

Let A = [aij ] and t ∈ F (that is t is a scalar).

Then t A is the matrix obtained by multiplyingall elements of A by t ; that is

t A = t [aij ] = [t aij ]:

Example

⎡1 −2⎤ ⎡3(1) 3(−2)⎤

3⎢⎣4 2 ⎥⎦ = ⎢⎣3(4) 3(2) ⎥⎦



Subtraction of matrices

Matrix subtraction is defined for two matrices

A = [aij ] and B = [bij ] of the same size; that is

A - B = [aij ] - [bij ] = [aij - bij ]

Example:

⎡1 2⎤ ⎡3 −1⎤ ⎡1−3 2+1⎤ ⎡−2 3⎤

⎢⎣4 5⎥⎦−⎢⎣1 4⎥⎦=⎢⎣4−1 5−4⎥⎦=⎢⎣3 1⎥⎦



Multiplication of Matrices

If A is an mxn matrix and B is an nxp matrix, then their matrix

product AB is the mxp matrix whose entries are given by dot product ofthe corresponding row of A and the corresponding column of B:



Example : ⎡1 3⎤ ⎡ 0 −1 1⎤

Let A=⎢⎣0 2⎥⎦ andB=⎢⎣− 2 5 4⎥⎦

,

where 1 ≤ i ≤ m and 1 ≤ j ≤ p.

∑

=

= + + + =

n

r

rj ir nj in i i B

1

2 2 1 1 ...



Solution :

⎣⎢⎡10 23

⎥⎦⎤⎡

⎢⎣−02 −51 14

⎥⎦⎤

=⎢⎡⎢⎣⎢[[01 32]]⎢⎡⎣⎡⎢−−0022⎥⎦⎤⎤⎥⎦ [[01

32]]⎡⎢⎣⎣⎡⎢−−5511⎥⎦⎤⎦⎥⎤ [[01 32]]⎣⎢⎡⎢⎣⎡1414⎤⎥⎦⎥⎤⎦⎥⎥⎥⎦⎤⎥=⎢⎡⎣−−64 1410 138⎤⎥⎦

⎢

⎣

Zero matrix



For each m; n the matrix in Mmxn(F), all of whose

elements are zero, is called the zero matrix

denoted by the symbol 0.

Example : The zero matrix of order

2 :

Identity matrix of order n

Is a matrix whose diagonal entries are 1 and allother entries zero is called the identity matrix of

order n denoted by In.

Examples :

⎥

⎦

⎤

⎢

⎣

⎡

=

1 0

0 1

2

⎥

⎥ ⎥

⎦

⎤

⎢

⎢ ⎢

⎣

⎡

=

1 0 0

0 1 0

0 0 1

3

⎥

⎦

⎤

⎢

⎣

⎡

0 0

0 0



Square matrices

A square matrix is a matrix with the same number of

rows and columns. An nxn matrix is known as a squarematrix of order n.

A square matrix A is called invertible or non-singular if

there exists a matrix B such that

AB = BA =In.

Moreover, if B exists, it is unique and is called the

inverse of matrix A, denoted A−1.

Example: Show that B is the inverse of A.

⎡−1 2⎤ ⎡1 −2⎤

Let A = ⎢−1 1⎥⎦ and B = ⎢⎣1 −1⎥⎦



⎣

Solution:

⎡−1 2⎤ ⎡1 −2⎤ ⎡1 0⎤

AB = ⎢⎣−1 1⎥⎦ ⎢⎣1 −1⎥⎦ = ⎢⎣0 1⎥⎦ and

⎡1 −2⎤⎡−1 2⎤ ⎡1 0⎤

BA = ⎢⎣1 −1⎥⎦⎢⎣−1 1⎥⎦ = ⎢⎣0 1⎥⎦

Since AB=BA=In, then B is the inverse of A.

Determinants



Definition 9:

The determinant of a matrix is the real

number associated to a given matrix A.

Determinant: Matrix of order 2



The area of the parallelogram is the absolute value of the determinant

of the matrix formed by the vectors representing the parallelogram's

sides.

Determinant: Matrix oforder 3



The volume of this Parallelepiped is the

absolute value of the determinant of the

matrix formed by r 1, r 2, and r 3.

Rule of Sarrus or the Diagonal Method



The rule of Sarrus is a

mnemonic for this formula: the

sum of the products of three

diagonal north-west to southeast

lines of matrix elements, minusthe sum of the products of three

diagonal south-west to north-east

lines of elements when the copies

of the first two columns of the

matrix are written beside it as in

the illustration at the right.

can be calculated by its diagonals.

Example: Find the determinant of the matrix A

The determinant of a 3x3 matrix





Cofactor Expansion

Let A be a square matrix of order n. Then thedeterminant of A is given by

Where the cofactor Cij is given by

Cij =(−1)i+ j

Mij

in in 3 i 3 i 2 i 2 i C a ... C a C a C a C a A ) A det( 1 i 1 i

n

1

ij ij + + + + = = = ∑

=



and where the minor Mij is the of the element aij is the

determinant matrix obtained by deleting the ith row

and the jth column of A.Example: Find the determinant of the matrix A

using cofactor expansion of the 1st row.



⎡0 2 1⎤

A =⎢⎢3 −1 2

⎥⎥ ⎢⎣4

0 1⎥⎦

Solution: Using the cofactor expansion of the

1st row:



Recall: 1+1−1 2

0 2 1⎤ C11 = (−1)0

⎡

A =⎢⎢⎢⎣34 −01 21

⎥⎥⎥⎦

C12 = (−1)1+

234

C13 = (−1)1+3⎡⎢⎣34

Thus we have

= −1,

1

2

= 5, and

1

−1⎤

0 ⎥⎦ = 4



A =a11C11+ a12C12 +

a13C13

A =0(−1)+2(5)+1(4)=14



Gabriel Cramer (1704-1752)

A Swiss mathematician who was a professor

of mathematics at Geneva. When Cramerpublished his rule in 1750 he did not use

determinants as they are now shown, and he

gave no explanation for how he achieved the

result. It seems that Colin Maclaurin probably

discovered the same rule as early as 1729, butit was not published until after his death. Although Cramer is

primarily remembered for the rule of determinants he also

worked in problems related to physics and general geometry

and algebraic curves.Solving Systems of Linear Equations



4.Cramer’s Rule



Solve the

system

2x + 9y = 8 x +5y =4

Solution:

(4,0)

Introduction : Gauss-

Jordan Elimination



Elementary Row Operations

1.Interchange two rows.

Ri ↔ R j

2. Multiply a row by a nonzero constant.

cRi ↔ Ri

3. Add a multiple of a row to another row.

R j+

cRi

↔ R j



Reduced Row-Echelon FormDefinition 9: A matrix is in reduced row-echelon form (RREF) 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 the leading 1.

3.The leading 1 of each row after the first is positioned to theright of the leading 1 of the previous row.

4.All other elements in a column that contains a leading 1 are

zero.

Examples:



⎡0 1 0 5⎤

A=⎢⎢0 0 1 3⎥⎥ Matrix A is in RREF.

⎢⎣0 0 0 0⎥⎦

⎡1 2 −3 4 ⎤ Matrix B is not in B=⎢⎢0 2 1

−1⎥⎥ RREF.

⎢⎣0 0 1 −3⎥⎦






Example:

Solve the system

x + y = 10

2 x +y = 15

1 1 101 1 101 1 10

−2 R1 + R2 → R2−1 R2 → R2

150 −1 −50 12 1 5

1 0 5

−1 R2 + R1 → R1

0 1 5

The system is reduced to x=5

y=5.



The solution is (5,5).

Introduction :Use of Inverse

Inverse of a matrix

When det(A)≠0, the inverse of a 2x2

matrix A is obtained using

⎥

⎦

⎤

⎢

⎣

⎡

−

−

=

−

a c

b d 1 1






5. Using the Inverse of a Matrix

THEOREM: Let AX=Y be a system of n linear equations in n variables. If A-

1 exists, the solution is unique and is given by X=A-1 Y.

Proof: First prove that X=A-1Y is a solution by substituting it into the matrix

equation. Using the properties of matrices we get

AX=A(A-1 Y)=(AA-1)Y=In Y=Y.

X=A-1Y satisfies the equation, thus it is a solution.

Now we prove the uniqueness of the solution. Let X1 be any solution. Thus

AX1=Y. Multiplying both sides of this equation by A-1 gives

A-1AX1=A-1 Y

InX1=A-1 Y

X1=A-1

Y

Thus there is a unique solution X1=A-1Y.



Linear Algebra w/Applications by Gareth

Williams

Example: x 1 + x2 = 10

2 x 1 + x

2 = 15

The system can be written in the following matrix form:

AX=Y ⎡1 1⎤⎡ x1⎤ ⎡10⎤ ⎢⎣2

1⎥⎦⎢⎣ x2⎥⎦=⎢⎣15⎥⎦

The inverse of the coefficient matrix is ⎡−1 1 ⎤

Applying the theorem A-1:

⎢

⎣ 2 −1

⎥

⎦

⎡ x1⎤ ⎡−1 1⎤⎡10⎤ ⎡5⎤



X=A-1Y ⎢ x2⎥⎦=⎢⎣2 −1⎥⎦⎢⎣15⎥⎦=⎢⎣5⎥⎦ ⎣

Thus the solution is x1 =5 andx2 =5

Let’s makeequations alive!

Some Applications



Curve Fitting

Networks

Cryptography



Application of Systems

of Equations toGeometry

Curve Fitting

1. Find a polynomial whose graph passesthrough the points. The points are oftenmeasurements in an experiment.

2. The x-coordinates are called base points. Itcan be shown that if the base points are all



distinct, then a unique polynomial of degreen-1 (or less) can be fitted to the points.



Example 1 : Determine the equationof the polynomial of degree two

whose graph passes through the

points (1,6),(2,3) and (3,2).

We want to find an equation of degree two.

(one less than the number of data points)

Let the polynomial be



y= a0 +a1 x+a2 x2

Now we solve for the values of

a0 , a1 , and a2



Substituting the points in the polynomial leads to

the following system of linear equations

a 0 + a1 + a 2 = 6 a 0 + 2a 2 +

4a 3 = 3 a 0 + 3a 2 + 9a 2 =

2

and using Gauss – Jordan Elimination gives

y=11−6 x+ x2

Recall: The pts.are (1,6),(2,3)

and (3,2).



Example 2 : Given three points



A1 =(x1, y1), A2 =(x2, y2) and A3 =(x3, y3) in the

plane (and not on the same line), find the

equation of the circle going through these points.







If M =(x, y) is an arbitrary point on the



circle, then we can write

where a, b, c and d are constants, a ≠ 0



Find the equation of the circle through the

points A (1 , 0), A (-1 , 2) and A (3 , 1). we



which gives after simplification

• The circle has (7/6, 13/6) as centerand 37 /18 as radius.





What is a NETWORK?

a network is a series of points or nodes

interconnected by communication paths.

Examples:

1.Street (Traffic) network

2.Circuit network



STREET

(TRAFFIC)NETWORK



Assumptions:

1.Assume that the streets are one way.

2. By Kirchhoff’s first Law, the flow into

an intersection is equal to the flow out.

The following diagram shows part of the central section of a

campus. Find the amount of the traffic between each of four

intersection .



120

70

A

B

D

C



For each intersection, this fact can be shown by an

A: x 4+120 = x 1 + 250

B: x 3 + 115 = x 4 + 175

C: x 2 + 630 = x 3 + 390

equation.

A

B

D

C

70

120



D: x 1 + 70 = x 2 + 120

Rewriting this system of linear

equations:

The augmented matrix of this system is



RREF

as a linear system :



A free variable exists, this problem has many possible

solutions, but x 4 > 180.



Electric

Circuit

A simple Electric

Circuit is a closed

connection of

batteries, resistors,and Wires. It consists

of voltage, loops and

current nodes.

The following physical

quantities are measured in anelectrical circuit; Current,: Denoted by I measured in Amperes (A).

Resistance ,: Denoted by R measured in Ohms ( W ) .



Electrical Potential Difference ,: Denoted by V measured in volts. (v)

Three basic laws governing the flow of current in an electrical circuit

1. Ohm's Law The voltage across the conductor is equal to theproduct of the resistance and the current flowing through it (at

constant temperature) : V=IR

2. Kirchhoff's Voltage Law The algebraic sum of the voltage drops

around a closed loop is equal to the total voltage in the loop.

3. Kirchhoff's Current Law The sum of all currents entering a node

is equal to the sum of all currents leaving the node.

Example Determine the currents I1, I2, and I3

for the following electrical network:

A



I1 I1

8 volts

2 ohms 2 ohms

I3 I3

B D

1 ohm

I2 I2

C

4 ohms 16 volts

Applying Kirchhoff’s first Law to either of the nodes B

or D,

I 2 + I 1 = I 3 or



I 1 + I 2 − I 3 =0

Applying Kirchhoff’s second Law to the loops ABDA and

CBDC, we obtain the equations

A

I1 I1

8 volts

2 ohms 2 ohms

I3 I3

B D

1 ohm

I2 I2

C

4 ohms 16 volts

2 I 1 +1 I 3 +2 I 1 = 8

4 I 2 +1 I 3 =16

Thi i th li t



This gives the linear system:

I 1 + I 2 − I 3 = 0

4 I 1 + I 3 = 8

4 I 2 + I 3 = 16

Whose augmented matrix is

⎡1 1 −1 0⎤

⎢⎢4 0 1 8

⎥⎥

⎢⎣0 4 1 16⎥⎦

RREF





Cryptography



Cryptography is the study of encoding

and decoding secret messages.

Codes are called ciphersplaintext - uncoded messages

ciphertext -coded messages

Encoding- converting a plaintext to a ciphertext

Decoding- converting a ciphertext to a

plaintext



Different ways of coding:

1. Substitution replace each letter of alphabet by a

different letter or a number.

For example, replace a with m, and b with k.

2. Polygraphic System

divide plain text into sets of n-letters, and

replace them with n code letters. In this case

invertible matrices can be used to provide a better



invertible matrices can be used to provide a better

coding, than substitution.

For example, we may assign an integer to each of

the letters of the English alphabet.

E l 1 E d h



Example1: Encode the message

“MATH IS MAGICAL”

using the encoding matrix

⎡4 −3⎤

A=⎢⎣3 −2⎥⎦



Using the conversion table below:

M A T H I S M A G I C A L

13 1 20 8 27 9 19 27 13 1 7 9 3 1 12

We assign 27 for every space between words.

RECALL:





13 1 20 8 27 9 19 27 13 1 7 9 3 1 12

divide the number into groups of 2x1 matrices

⎡13⎤ ⎡20⎤ ⎡27⎤ ⎡19⎤ ⎡13⎤ ⎡7⎤ ⎡3⎤ ⎡12⎤

⎢⎣ 1 ⎦⎥,⎢⎣ 8 ⎥⎦,⎢⎣ 9 ⎥⎦,⎢⎣27⎥⎦,⎣⎢ 1 ⎦⎥,⎢⎣9⎥⎦,⎢⎣1⎥⎦,⎢⎣27⎥⎦

Use these columns as the columns of the matrix:

⎡13202719137 312⎤



⎢⎣1 8 9 271 9 1 27

⎥⎦

We multiply the matrix obtained to the encodingmatrix to obtain the encoded message(ciphertext).

⎡4 −3⎤⎡1320 2719137 3 12⎤ ⎡ ⎤

⎢⎣3 −2

⎥⎦⎢⎣1 8 9 27 1 9 1 27⎦

⎥=⎢⎣

⎥⎦

Thus the encoded message is:

E l 2 D d th



Example2: Decode the message

__________________

which was encoded using the

matrix

⎡4 −3⎤ A=⎢⎣3−2⎥⎦

Solution: First we have to find the decoding matrix



Solution: First we have to find the decoding matrix ,

or the inverse of⎡4 −3⎤ A=⎢⎣3

−2⎥⎦

RECALL: −1 = A1 ⎡⎢⎣−dc −ab⎤⎥⎦

A

Thus,

⎢⎣ ⎥ ⎦ ⎢⎣ ⎥⎦



A−1 = −81+9 ⎡⎢⎣−−23 34⎥⎤⎦=⎢⎣⎡−−23 34⎤⎥⎦

This matrix will serve as the decoding matrix.

⎡ ⎤

⎢ ⎥

⎡⎢−−23 34⎤⎥⎦[ ]=⎢⎢⎢ ⎥⎥⎥

⎣

⎢ ⎥ ⎢⎣ ⎥⎦

The message is



The message is

Try this!



Try this!

Decode the message

Which was encoded using the matrix



Workshop!

11:00-11:20 Group discussions

11:20-11:30 Group reports

1. What are the important concepts students should be equipped

with before studying systems of linear algebra.

2. How can we motivate students to study systems of linear

equations?

3. What strategies have you used that you found effective in

teaching systems of equations?

References:

Linear Algebra with Applications by Gareth Williams

Elementary Linear Algebra by Ron Larson and David C Falvo



Elementary Linear Algebra by Ron Larson and David C. Falvo

http://aix1.uottawa.ca/~jkhoury/networks.htm

http://www.math.ucdavis.edu/~daddel/linear_algebra_appl/Applications/applications.html



Thank you for listening!

Date post:	05-Mar-2016
Category:	Documents
Upload:	elizabeth-solayao
View:	4 times
Download:	0 times

System of Linear Equation

Documents