LINEAR ALGEBRA.pptx

8/10/2019 LINEAR ALGEBRA.pptx

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The most practical applications of mathematics are

those that apply to the most people.

Whether it is data through traffic, arranging anefficient flow is a crucial element of design.

Many of the problems that arise with networksand traffic involve optimizing the flow through thesystem.

However, by looking at systems of linear equations,

we can simplify flow problems by studying thesesystems using matrices and determine whichvariables have forced values and which are free tobe chosen.


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come up with equations

one equations for each node incoming traffic or flow needs to equal the outgoing

traffic or flowStep1

rearrange these equations in order get the constant on one side and all of the variables

on the other sideStep

2

create a matrix

each column of the matrix corresponds to one variable

final column will contain the constants from eachequation

Step3


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row-reduce ,(REF)the matrix accomplished by using the 3 elementary

row operationsStep4

convert back into linear equation

reverse process of step 3Step

5


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Example 1

40 30

50 B A

C E

70 D 25


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Step 1 - come up with equations

B + E = 30 + A

C + 40 = B + 50

70 = C + D

D = E + 25


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Step 2 - rearrange these equations in order

-A + B + E = 30

-B + C = 10

C + D = 70

D

E = 25

Step 3 - create a matrix

-1 1 0 0 1 30

0 -1 1 0 0 10

0 0 1 1 0 70

0 0 0 1 -1 25


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Step 4 - row-reduce ,(REF) the matrix

1 0 0 0 0 5

0 1 0 0 1 35

0 0 1 0 1 45

0 0 0 1 -1 25

Step 5 - convert back into linear equation

A = 5

B + E = 35

C + E = 45

DE = 25


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From these equations (step 5), we can see that A

must be 5 cars per hour. But for they have some choice for the other

streets.

If E = 20 ;

B = 15C = 25

D = 5

If E = 0 ;

B = 35C = 45

D = 25


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When looking at the application , data throughnetwork traffic, the system can be set up similarly.

The reason that linear algebra is so key of solving

these types of problems - that it allows for solvingmany linear equations simultaneously.

Matrices use the symmetry of the equations to

solve for the variable.


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LINEAR ALGEBRA.pptx

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