2MA101-Linear Algebra

Depar tment of Mathematics & Humanities,

Institute of Technology,

Nirma University

B. Tech. - Semester I

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Mathematics in Engineering

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• Linear Algebra• Calculus• Differential Equations• Special Functions• Complex Functions• Advance Transformation• Advance Mathematical Techniques• Optimization Techniques…

At the heart of engineering is an understanding of engineering design processes andthe creation, application, and adaptation of mathematical/scientific models thatcan be used to interpret, explain, and predict the behaviour of complex systems.

- Cunningham & Hester, 2007 A basic engineering design process involves the following cyclic components:

ASK—What is the problem? What have others done? What are the constraints?

IMAGINE—What are some possible solutions? Brainstorm ideas. Choose the bestone.

PLAN—What diagram can we draw to help us here? Make a list of materialsneeded.

CREATE—Follow your plan and create it. Test it out.

IMPROVE—Discuss what works, what doesn’t, and what could work better.Modify your design to make it better, Test it out. 3

Solve : Answer :

x1 = 0.2,

x2 = 0.4,

x3 = 0.4.

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1 3

2 1 3

3 2

0.5

0.5

x x

x x x

x x

& x1 + x2 + x3 = 1

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•Why?

Simple clean design, fast searches, it puts the “good stuff”

at the top

• Some Figures

Google indexes billions of pages

It has several hundred million queries per day with many,

many thousands per second

It runs on tens of thousands of Linux computers

It makes up half of the multi billion dollar per year search

engine business

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Google has to do three things

a) Crawl the web to find all the publicly

accessible websites

b) Arrange the data it finds so that it can be

searched quickly

c) Rank the websites in order of

importance, so that the most important

can be presented to the user first

The rest of this presentation focuses on c) above.

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Webs

ite

3

Webs

ite

2

Webs

ite

1

Suppose the entire world wide web consisted of only three

websites. An arrow between two websites in the diagram

on the left represents a link between those two websites.

For example there is a link from website 2 to website 3.We

can represent this as a directed graph, see the diagram on

the right.

2

3

1

Website Linked to

from

1 3

2 1,3

3 2

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2

3

1

The goal is to assign a rank to

each website.

The rank should be a measure

of interest.

Let x1 be the rank of website 1

Let x2 be the rank of website 2

Let x3 be the rank of website 3

We’ll also insist that the ranks

are normalised,

i.e. x1 + x2 + x3 = 1

First attempt is to base the rank

on the number of incoming links.

Website

i

Incoming

LinksRank, xi

1 1 0.25

2 2 0.50

3 1 0.259

3

1

For example website 3 has voting power of

x3.

Since it links to websites 1 and 2, it will vote

0.5x3 to website 1 and 0.5x3 to website 2.

Website 1 has voting power of x1. Since it

only links to website 2, it will vote x1 to

website 2.

Website 2 has voting power of x2. Since it only

links to website 3, it will vote x2 to website 3.

But the total of each website’s vote is its

rank. This gives the following

equations…… 10

1 3

2 1 3

3 2

0.5

0.5

x x

x x x

x x

2

31

But this can be written as a matrix equation….

1 1

2 2

3 3

0 0 0.5

1 0 0.5

0 1 0

x x

x x

x x

The solution of this

with

x1 + x2 + x3 = 1 is

x1 = 0.2, x2 = 0.4, x3 =

0.411/10/2020 11

2

3

1

1 1

2 2

3 3

0 0 0.5

1 0 0.5

0 1 0

x x

x x

x x

The ranks x1, x2, x3 are a

solution of

So the ranks as a column vector are an eigenvector of

the matrix with Eigen value 1 (chosen so that the

ranks add up to 1). You might know this as a fixed

point of the matrix.

How does Google calculate this so quickly? We need

to find a quick way to calculate this vector, and hence

the ranks.

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0 0 0.5

1 0 0.5

0 1 0

T

2

31

Let

Starting with the vector

1/ 3

1/ 3

1/ 3

x

We look at the sequence of

vectors Tx , T2x, T3x,

T4x,…..

The limit of this sequence is

the eigenvector of ranks….

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0 0 0.5 0

1 0 0.5 1

0 1 0 0

0 0 0 0

T

2

3

1

Let

4

Site 1 2 3 4

1 0 0 1 0

2 1 0 1 1

3 0 1 0 0

4 0 0 0 0

gives ranks 0.2, 0.4, 0.4, 0

respectively

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0 0 0.5 0

1 0 0.5 1

0 0.5 0 0

0 0.5 0 0

T

2

31

Let

4Sit

e1 2 3 4

1 0 0 1 0

2 1 0 1 1

3 0 1 0 0

4 0 1 0 0

gives ranks 1/9, 4/9, 2/9, 2/9

respectively

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Google uses simultaneous equations in anextremely inventive way to do its page ranking.This made it the market leader in search engines!

What is the next big idea in applications ofmathematics?.............

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What is Linear Algebra?

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Matrix Theory

Vector Space & Linear Transformation

Eigen Value, Eigen Vector & ApplicationsINTERSE WEIGHTAGE (CE/LPW)

Semester: B. Tech. Semester – I Duration: October, 2020 -February, 2021

COMPONENTS WEIGHTAGE

Branch

Subject

code and

name

Course

Coordinator

Faculties

InvolvedL T P C

Continuous Evaluation (CE) LPW

Class

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Practical

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2MA101-

Linear

Algebra

Dr. Amit

Mishra

Dr. Sandeep

Malhotra 3 1 - 4 0.30 0.40 0.30 - - -

Course Blog: ma202linearalgebra@wordpress.com

Linear Algebra (2MA101)

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Reference Books:1. Lay, D. C. (2015) Linear Algebra and its Application. New Delhi:

Pearson Publication

2. Kreyszig, E. (2006) Advanced Engineering Mathematics. Singapore:John Wiley Publication

3. Anton H.(2005) Elementary linear algebra with applications, NewDelhi: John Wiley Publication

4. Hoffman, K. & Kunze, R. (1971) Linear Algebra. New Delhi: PHIPublication

5. Kumaresan, S. (2011) Linear algebra - A Geometric approach, NewDelhi: PHI Publication

6. Sharma, J. P. & Yeolekar, M. (2011) Engineering mathematics,Vol-II. New Delhi: PHI Publication11/10/2020 DR. SANDEEP MALHOTRA 19

Applications of Linear Algebra

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• Electric Circuit Analysis• Control Systems• Artificial Intelligence• Machine learning• Robotics Moments• Heat Transfer Analysis• Transport Phenomena• Construction Technology (Finite Element Analysis)• Communication Networks…

Linear Algebra in Electrical CircuitWe want to know the currents in the following Circuit

problem!!

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The circuit equations, using Krichhoff's Law:

-26 = 72I1 - 17I3 - 35I4

34 = 122I2 - 35I3 - 87I7

-13 = 149I3 - 17I1 - 35I2 - 28I5 - 35I6 - 34I7

5 = 105I4 - 35I1 - 43I5

-27 = 105I5 - 28I3 - 43I4 - 34I6

24 = 141I6 - 35I3 - 34I5 - 72I7

-4 = 233I7 - 87I2 - 34I3 - 72I6

Hard one??!!!

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We try to use matrices to solve it in easy way!!!

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And the answers will be.....

I1 = -0.4680 A

I2 = 0.4293 A

I3 = 0.0005 A

I4 = -0.2224 A

I5 = -0.2785 A

I6 = 0.2112 A

I7 = 0.2091 A

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The Use of Linear Algebra

in Space Shuttle Control Systems

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Rotations in Three Space:

The rotation matrix for the 3*3 case is defined as follows:

- Rotation Around the x-axis:

A roll is a counter clockwise rotation of gamma about the x-axis;

the rotation matrix is given by:

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- Rotation Around the y-axis:

A pitch is a counter clockwise rotation of beta about the y-axis; the

rotation matrix is given by:

- Rotation Around the z-axis:

A Yaw is a counter clockwise rotation of alpha about the z-axis; the

rotation matrix is given by:

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In the picture below, the figure at left is first rotated 90 degrees about the z-

axis, then rotated 30 degrees about the x-axis. The result is shown in the top

right figure. Next, the same operations are performed, but in opposite order,

as shown in the bottom right view. It is now obvious that these operations

are not commutative.

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Putting it All Together:

Q: What does this Mean for the space Shuttle?

One can now expand these principles to derive a sequence of control

operations to use for directing 3-dimensional objects such as the space

shuttle. This is one of the greatest benefits to be had from the application of

Linear Algebra.

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“The book of the Universe liescontinually open to man’s gaze, yetnone can hope to comprehend itwho has not first mastered thelanguage. This language ismathematics.” - Galileo

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