An Introduction to Vector Spaces

Abstract

The theory of matrices developed in Chapters 1–6, and the algebra of vectors in three dimensions used throughout calculus and physics, hereafter called space vectors, both belong to the part of mathematics called linear algebra. Each is a particular example of a linear algebra, with matrices being the more general of the two. At first sight the algebra of matrices and of space vectors appear be very different, but this is due to the use of different notations when describing vectors themselves, and the operations of vector addition and the scaling of vectors by a real number λ. General space vectors r = a i + b j + c k are constructed by the scaling and addition of the unit vectors i, j and k that are parallel to the orthogonal x, y and z-axes, and thereafter the algebra of space vectors is developed in terms of these unit vectors. However, vector r with its components a, b and c can equally well be defined as a three element row or column matrix, after which the linear operations of the scaling and addition of matrix vectors can be developed using the rules of matrix algebra.