Vector Spaces

Abstract

We now begin our treatment of the principal subject matter of this text. We shall see that all of linear algebra is essentially a study of various transformation properties defined on a vector space, and hence it is only natural that we carefully define vector spaces. This chapter therefore presents a fairly rigorous development of (finite-dimensional) vector spaces, and a discussion of their most important fundamental properties. Basically, the general definition of a vector space is simply an axiomatization of the elementary properties of ordinary three-dimensional Euclidean space. A nonempty set V is said to be a vector space over a field F if: (i) there exists an operation called addition that associates to each pair x, y ∞ V a new vector x + y ∞ V called the sum of x and y; (ii) there exists an operation called scalar multiplication that associates to each a ∞ F and x ∞ V a new vector ax ∞ V called the product of a and x; (iii) these operations satisfy the following axioms: (V1) x + y = y + x for all x, y ∞ V. (V2) (x + y) + z = x + (y + z) for all x, y, z ∞ V. (V3) There exists an element 0 ∞ V such that 0 + x = x for all x ∞ V.