An Introduction to Linear Algebra

This book is meant for an introductory course in linear algebra for undergraduate students of mathematics. It deals with the concept of vector spaces and special types of functions defined on them called linear transformations or operators. The vector spaces considered in the book are finite-dimensional, a concept that involves representation of vectors in terms of a finite number of vectors which form a basis for the vector spaces. Written from a students perspective, this textbook explains the basic concepts in a manner that the student would be able to grasp the subject easily. Numerous solved examples and exercises given at the end of nearly each section will help the student to gain confidence in his/her analytical skills. What makes this book probably stand apart from other standard books on finite-dimensional linear algebra is the introduction to Hilbert Space Theory. The generic model of a finite-dimensional Hilbert space (real or complex) is IRn or sn but the true relevance of operators in Hilbert spaces surfaces only when they are infinite-dimensional. In order to properly comprehend the structure of an infinite-dimensional Hilbert space, it is important to grasp it at the finite-dimensional level. Although finite-dimensional Hilbert spaces are discussed comprehensively in the first eight chapters, it is only in the last three chapters that the treatment of Hilbert spaces is given in a setting which can be easily extended to defining infinite-dimensional Hilbert spaces. After going through this textbook, the students will have a clear understanding of the model of a Hilbert space in finite-dimensions and will then be able to smoothly make the transition to infinite-dimensional Hilbert Space Theory.