Application Of Linear Algebra

Linear Algebra is used in various fields of Science and Technology. 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…. 1.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. 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. 2. 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. 3. 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. This law when used with Hooke’s Second Law helps to find the oscillations of coupled springs arranged in various examples 4. 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. 5. 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. It is used in ATM cards ,MOBILE passwords, COMPUTER locks, SUPER CARDS and MOBILE cards etc. 6.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). 7.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 . 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. 9. 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. Distances, Eq of st: lines ,direction ratios, direction QIBLA ,Eq of panes etc