This book introduces some of the elementary concepts and results of Linear Algebra. It explains basic concepts and techniques of linear algebra and make them accessible to the undergraduate students. The fundamental concepts of Rings, Integral domains, Fields, Ideals, Quotient Rings, Homomorphism of Rings, Polynomial Rings, Systems of Linear equations, Vector Spaces, Linear Transformations, Vector Space Isomorphism, Inner Product Spaces and Real Quadratic forms are discussed. Each chapter includes clear statements of pertinent definitions, principles and theorems together with illustrative and descriptive material.