This document discusses various applications of linear algebra in different fields such as abstract thinking, chemistry, coding theory, cryptography, economics, elimination theory, games, genetics, geometry, graph theory, heat distribution, image compression, linear programming, Markov chains, networking, and sociology. It provides examples of how linear algebra concepts such as systems of linear equations and matrix operations are used in topics like balancing chemical equations, error detection in coding, encryption/decryption, economic models, genetic inheritance, and finding lines and circles in geometry.

1. Applications of Linear
Algebra
A Group I Project By :
Nirav Patel - 140110111041
Parth Patel - 140110111042
Vishal Patel -140110111043
Prerak Trivedi - 140110111045
Prutha Parmar - 140110111046
Tanvi Ray - 140110111048

2. Linear Algebra
› What is Linear Algebra?
› Applications of Linear Algebra in various fields.
– Abstract Thinking
– Chemistry
– Coding Theory
– Cryptography
– Economics
– Elimination Theory
– Games
– Genetics
– Geometry
– Graph Theory
– Heat Distribution
– Image Compression
– Linear Programming
– Markov Chains
– Networking
– Sociology
– The Fibonacci Numbers
– Eigenfaces
and many more….

3. What is Linear Algebra?
› Linear Algebra is the branch of mathematics concerning
vector spaces and linear mappings between such spaces.
It includes the study of lines, planes, and subspaces, but
is also concerned with properties common to all vector
spaces.
› Hence, the above definition confirms that Linear Algebra
is an integral part of mathematics.

4. Abstract Thinking
› Linear Algebra has over some other subjects for
introducing abstract thinking, is that much of the material
has a geometric interpretation.
› In low dimensions, one can "visualize" algebraic results,
and happily, the converse is also true: linear algebra helps
develop your geometric instinct.
› The geometric intuition you already have will be
complemented by an "algebraic picture", one that will
allow you, with practice, to "see" in higher dimensions that
are inaccessible to our normal senses.

5. Chemical Applications
› Application of linear systems to chemistry is balancing a
chemical equation and also finding the volume of
substance. The rationale behind this is the Law of
conservation of mass which states the following:
› “Mass is neither created nor destroyed in any chemical
reaction. Therefore balancing of equations requires the
same number of atoms on both sides of a chemical
reaction. The mass of all the reactants (the substances
going into a reaction) must equal the mass of the
products (the substances produced by the reaction).”

6. As an example consider the following chemical equation
C2H6 + O2 → CO2 + H2O.
Balancing this chemical reaction means finding values of x, y, z and t so that
the number of atoms of each element is the same on both sides of the
equation:
xC2H6 + yO2 → zCO2 + tH2O.
This gives the following linear system:
The general solution of the above system is:
Since we are looking for whole values of the variables x, y z, and t, choose x=2
and get y=7, z= 4 and t=6. The balanced equation is then:
2C2H 6 + 7O2 → 4CO2 + 6H2O.

7. Applications in Coding Theory
Transmitted messages, like data from a satellite, are always subject to
noise. It is important; therefore, to be able to encode a message in such a
way that after noise scrambles it, it can be decoded to its original form.
This is done sometimes by repeating the message two or three times,
something very common in human speech. However, copying data stored
on a compact disk, or a floppy disk once or twice requires extra space to
store.
In this application, we will examine ways of decoding a message after it
gets distorted by some kind of noise. This process is called coding. A code
that detects errors in a scrambled message is called error detecting. If, in
addition, it can correct the error it is called error correcting. It is much
harder to find error correcting than error-detecting codes

8. Most messages are sent as digital-sequences of 0’s and 1’s, such as 10101 or
1010011, so let us assume we want to send the message 1011. This binary “word”
may stand for a real word, such as buy, or a sentence such as buy stocks.
One way of encoding 1011 would be to attach a binary “tail” to it so that if the
message gets distorted to, say, 0011, we can detect the error. One such tail could be
a 1 or 0, depending on whether we have an odd or an even number of 1’s in the word.
This way all encoded words will have an even number of 1’s. So 1011 will be encoded
as 10111.
Now if this is distorted to 00111 we know that an error has occurred, because we only
received an odd number of 1’s. This error-detecting code is called a parity check and
is too simple to be very useful.
For example, if two digits were changed, our scheme will not detect the error, so this
is definitely not an error-correcting code. Another approach would be to encode the
message by repeating it twice, such as 10111011.
Then if 00111011 were received, we know that one of the two equal halves was
distorted. If only one error occurred, then it is clearly at position 1. This coding
scheme also gives poor results and is not often used. We could get better results by
repeating the message several times, but that takes space and time.

9. Coupled Oscillations
› Everyone unconsciously knows this Law. Everyone knows that heavier
objects require more force to move the same distance than do lighter
objects. The Second Law, however, gives us an exact relationship
between force, mass, and acceleration:
› In the presence of external forces, an object experiences an
acceleration directly proportional to the net external force and inversely
proportional to the mass of the object.
› This Law Is widely known with the following equation:
F=ma

10. › The above Newton’s Second Law when used with Hooke’s Second Law
helps to find the oscillations of coupled springs arranged in various
examples.

11. › Cryptography, to most people, is concerned with keeping
communications private. Indeed, the protection of sensitive
communications has been the emphasis of cryptography
throughout much of its history.
› Encryption is the transformation of data into some unreadable
form. Its purpose is to ensure privacy by keeping the information
hidden from anyone for whom it is not intended, even those who
can see the encrypted data. Decryption is the reverse of
encryption; it is the transformation of encrypted data back into
some intelligible form.
CRYPTOGRAPHY

12. Encryption and decryption require the use of some secret information,
usually referred to as a key. Depending on the encryption mechanism
used, the same key might be used for both encryption and decryption,
while for other mechanisms, the keys used for encryption and
decryption might be different.
Today governments use sophisticated methods of coding and decoding
messages. One type of code, which is extremely difficult to break,
makes use of a large matrix to encode a message. The receiver of the
message decodes it using the inverse of the matrix. This first matrix is
called the encoding matrix and its inverse is called the decoding
matrix.

13. › Example Let the message be
PREPARE TO NEGOTIATE
and the encoding matrix be
We assign a number for each letter of the alphabet. For simplicity, let us
associate each letter with its position in the alphabet: A is 1, B is 2, and
so on. Also, we assign the number 27 (remember we have only 26 letters
in the alphabet) to a space between two words. Thus the message
becomes:

14. › Solving the above matrix in various ways, we can decrypt the
message as:
› So finally decrypting the message, we get :

15. Economics
› In order to understand and be able to manipulate the
economy of a country or a region, one needs to come up
with a certain model based on the various sectors of this
economy.
› The Leontief model is an attempt in this direction. Based
on the assumption that each industry in the economy has
two types of demands: external demand (from outside the
system) and internal demand (demand placed on one
industry by another in the same system), the Leontief
model represents the economy as a system of linear
equations.

16. › Consider an economy consisting of n interdependent
industries (or sectors) S1,…,Sn. That means that each
industry consumes some of the goods produced by the
other industries, including itself (for example, a power-
generating plant uses some of its own power for production).
› We say that such an economy is closed if it satisfies its own
needs; that is, no goods leave or enter the system. Let mij be
the number of units produced by industry Si and necessary
to produce one unit of industry Sj. If pk is the production
level of industry Sk, then mij pj represents the number of units
produced by industry Si and consumed by industry Sj .

17. › If pk is the production level of industry Sk, then mij pj represents the
number of units produced by industry Si and consumed by industry Sj.
Then the total number of units produced by industry Si is given by:
p1mi1+p2mi2+…+pnmin
In order to have a balanced economy, the total production of
each industry must be equal to its total consumption. This
gives the linear system:
If

18. › then the above system can be written as AP=P, where
A is called the input-output matrix.

19. Application to Elimination Theory
› Many problems in linear algebra (and many other branches of science) boil
down to solving a system of linear equations in a number of variables. This
in turn means finding common solutions to some “polynomial” equations of
degree 1 (hyperplanes).
› In many cases, we are faced with “nonlinear” system of polynomial
equations in more than one variable. Geometrically, this means finding
common points to some “surfaces”. Like the Gaussian elimination for linear
systems, the elimination theory in general is about eliminating a number of
unknowns from a system of polynomial equations in one or more variables
to get an easier equivalent system.
› One way of find common solutions to polynomial equations is to solve
each equation separately and then compare all the solutions. As you can
guess, this is not an efficient way especially if the goal is only to show the
existence of a solution.

20. › To understand the importance of elimination theory, let us start by considering the following example.
› Example 1 Without solving the polynomial equations, show that the following system has solutions.
Solution We compute the resultant of the two polynomials
therefore, the polynomials f(x), g(x) have a common root by the above theorem.
› One can use the above theorem to determine if a polynomial system in more than one variable has a
solution. The trick is to look at the polynomials in the system as polynomials in one variable with
coefficients polynomials in the other variables.

21. Applications in various GAMES
› GAME OF MAGIC SQUARES :
› A magic square of size n is an n by n square matrix whose entries
consist of all integers between 1 and n2, with the property that the
sum of the entries of each column, row, or diagonal is the same.
› The sum of the entries of any row, column, or diagonal, of a magic
square of size n is n(n2+1)/2 (to see this, use the identity:
1+2+...+k=k(k+1)/2).

23. Application to Genetics
› Living things inherit from their parents many of their physical
characteristics. The genes of the parents determine these
characteristics. The study of these genes is called Genetics; in
other words genetics is the branch of biology that deals with
heredity.
› In particular, population genetics is the branch of genetics that
studies the genetic structure of a certain population and seeks to
explain how transmission of genes changes from one generation to
another. Genes govern the inheritance of traits like sex, color of
the eyes, hair (for humans and animals), leaf shape and petal
color (for plants).
› There are several types of inheritance; one of particular interest
for us is the autosomal type in which each heritable trait is
assumed to be governed by a single gene. Typically, there are two
different forms of genes denoted by A and a.
› Each individual in a population carries a pair of genes; the pairs
are called the individual’s genotype. This gives three possible
genotypes for each inheritable trait: AA, Aa, and aa (aA is
genetically the same as Aa).

24. › in a certain animal population, an autosomal model of inheritance controls eye
coloration. Genotypes AA and Aa have brown eyes, while genotype aa has blue
eyes. The A gene is said to dominate the a gene. An animal is called dominant if
it has AA genes, hybrid with Aa genes, and recessive with aa genes. This means
that genotypes AA and Aa are indistinguishable in appearance.
› Each offspring inherits one gene from each parent in a random manner. Given the
genotypes of the parents, we can determine the probabilities of the genotype of
the offspring. Suppose that, in this animal population, the initial distribution of
genotypes is given by the vector
is called the transition matrix. In general, Xn=AXn-1. Explicitly, we have:

25. › Observe that the aa type disappears after the initial
generation and that the Aa type becomes a smaller and
smaller fraction of each successive generation. It is obvious
that this sequence of vectors converges to the vector
in the long run.
Now try to create a similar model for crossing offspring with
a hybrid animal. You will see that some offspring will have
brown eyes and some blue eyes.

26. GEOMETRICAL APPLICATIONS
› Given some fixed points in the plane or in 3-D space, many
problems require finding some geometric figures passing
through these points. The examples we are going to see in
this page require knowledge of solving linear systems and
computing determinants.
› Application 1 Let A1 = (x1, y1) and A2 =(x2, y2) be two fixed
points in the plane. Find the equation of the straight line L
going through A1 and A2.

27. › Solution Let M= (x, y) be an arbitrary point on L, then one
can find three constants a, b and c satisfying
Since A1 and A2 are on L, one has
Together with the above equation, we have a homogeneous
system in three equations and three variables a, b and c:
Since we know that there will be a line through A1 and A2;
this system will have at least one solution (a, b, c).

28. › However, if (a, b, c) is a solution, so is k(a, b, c) for any
scalar k and so the system has infinitely many solutions.
Therefore, the determinant of the coefficient matrix must be
zero.
or
For example if A1 =(-1, 2) and A2= =(0,1), then the equation of the line
L is in this case:

29. › Application 2 Given three points A1 =(x1, y1), A2 =(x2, y2)
and A3 =(x3, y3) in the plan (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

30. where a, b, c and d are constants. Substituting the three
points in the above equation gives the following homogeneous
system in four equations and four variables a, b, c and d:
As in Example 1, the system has infinitely many solutions. So,
the determinant:
For example, to find the equation of the circle going through
the points A1 (1, 0), A2 (-1, 2) and A3 (3, 1), we write

31. which gives after simplification
The circle has (7/6, 13/6) as center and 37/18 as radius.
This can also be used to calculate orbit of planets using
Kepler’s First Law.