Concepts In Vector Algebra

A vector V in the plane or in space is an arrow: it is determined by its length, denoted jVj and its direction. Two arrows represent the same vector if they have the same length and are parallel . We use vectors to represent entities which are described by magnitude and direction. For example, a force applied at a point is a vector: it is completely determined by the magnitude of the force and the direction in which it is applied. An object moving in space has, at any given time, a direction of motion, and a speed. This is represented by the velocity vector of the motion. More precisely, the velocity vector at a point is an arrow of length the speed (ds=dt), which lies on the tangent line to the trajectory. The success and importance of vector algebra derives from the interplay between geometric interpretation and algebraic calculation. In these notes, we will define the relevant concepts geometrically, and let this lead us to the algebraic formulation. The main structures of linear algebra are vector spaces. A vector space over a field F is a set V together with two binary operations. Elements of V are called vectors and elements of F are called scalars. This book offers a comprehensive description of the applications of various fields in this subject. The book will be appropriate as a guide for students.