This book is designed to teach the basic concepts of Linear Algebra precisely as a mathematical subject. Importance is given to definitions and theoremes. Complete and detail proof of everyt theorem is discussed. At least one example for each method is solved. This will develop confidence in the student to understand each topic. Reader must note that there is little or no emphasis on a geometric approach to problems of linear, the approach here is consciously algebraic. And this is one of the reasons to give title of book as "Elementary Linear Algebra".