Machine Learning and Big Data

1Mathematical Foundation

Afroz* and Basharat Hussain

Department of Mathematics, Maulana Azad National Urdu University, Hyderabad, India

Abstract

The aim of this chapter is to provide the reader an overview of basics of linear algebra and introductory lecture on calculus. We will discuss concept of real vector spaces, basis, span, and subspaces. The idea of solving the system of equations using matrix approach will be discuss. Linear transformation by means of which we can pass from one vector space to another, inverse linear transformation, and transformation matrix will be explain with detail examples. Definition of eigenvectors, eigenvalues, and eigendecomposition along with thorough examples will be provided. Moreover, definition of function, limit, continuity, and differentiability of function with illustrative examples will be included.

Keywords: Vector spaces, basis, linear transformation, transformation matrix, eigenvalue, eigenvector, eigen decomposition, continuous functions, differentiation

1.1 Concept of Linear Algebra

1.1.1 Introduction

Basics problem of linear algebra is to solve n linear equations in n unknowns.

For example,

The above system is two dimensional (n = 2), i.e., two equations with two unknowns. The solution of the above system is the values of unknowns x, y, satisfying the above linear system. One can easily verify that x = 1, y = 2 satisfy the above ...