APPLICATIONS OF LINEAR ALGEBRA IN INFORMATION TECHNOLOGY MADE BY : NIDHI SUTARIYA-17G055 PRIYANK MISTRY-17G050 UTSAV MODI-17G051 CHINMAY MISTRY-17G049 MUFADDAL NAYA-17G052 MUSAAB SHIRGAR17G053 NAYAN NATANI-17G054 URVISH THAKKAR-17G068

WHAT IS LINEAR ALGEBRA ? Linear algebra is a branch of mathematics concerning linear equations such as: a1x1 + a2x2 + …. + anxn = b, linear functions such as : (x1,x2 …. xn) -- a1x1 + a2x2 + …. + anxn and their representations though matrices and vectors

WHY LINEAR ALGEBRA IS IMPORTANT ? Linear algebra is vital in multiple areas of science in general. Because linear equations are so easy to solve, practically every area of modern science. It converts large number of problems to matrix and thus we can solve the matrix.

LINEAR ALGEBRA IN IT FIELD Linear algebra in Information Technology can broadly divided into two categories: Linear algebra for spatial quantities. Here you're dealing with 2-, 3-, or 4- dimensional vectors and you're concerned with rotations, projections, and other matrix operations that have some spatial interpretation. This is the kind of linear algebra that comes up, for example, in computer graphics or physics simulations. Linear algebra for statistics. Here you're dealing with vectors in high-dimensional spaces that have no particular spatial interpretation and you're interested in matrix decompositions and so on. This domain includes signal processing, statistical machine learning, and compression.

OBJECTIVES Importance of linear algebra Graph theory Network models Cryptography Computer graphics Matlab Programming

GRAPH THEORY

Q. Consider an undirected random graph of eight vertices Q. Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three? A cycle of length 3 can be formed with 3 vertices. There can be total 8C3 ways to pick 3 vertices from 8. The probability that there is an edge between two vertices is 1/2. So expected number of unordered cycles of length 3 = (8C3)*(1/2)^3 = 7

NETWORK MODELS This model was developed to overcome the problem of hierarchical model. The network model is a database model conceived as a flexible way of representing object and their relationship.

CRYPTOGRAPHY Encryption and decryption require the use of some secret information, usually referred to as a key. Example Let the message to be encrypted be: “PREPARE TO NEGOTIATE” We assign a number for each letter of the alphabet.

Thus the message becomes: Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1 vectors:

By multiplying encoding matrix to this matrix we will encrypt the msg

Now to decrypt the msg we have to multiply this matrix to Inverse of encoding matrix The inverse of this encoding matrix, the decoding matrix, is: Multiplying again by this matrix we will get our Msg.

COMPUTER GRAPHICS In computer graphics every element is represented by a MATRIX which are pixels. Computer graphic operates on Frames per second. For every frame the object to be displayed is stored in matrices.

Matlab Programming Matlab directly solves the standard linear programming problem using slack variables Matlab is all about Linear Algebra. One can perform any operation on matrices and derive the result very fast and efficiently.