CHAPTER V - VECTOR SPACES AND LINEAR EQUATIONS

Abstract

This chapter discusses vector spaces and linear equations. The chapter presents an assumption where K is an arbitrary number field. A sequence a = [a1, a2, …, an], consisting of n elements, where ak are elements of the field k, is called a vector of the space Kn and the set of these vectors—the vector space Kn over the field K. Numbers ak are called the components of the vector. The chapter presents a theorem which states that a sum of an arbitrary number of vectors depends neither on the arrangement of the terms nor on the manner of joining its terms in brackets. On changing the arrangement of terms or joining the terms in groups (indicated by the brackets), one gets a vector whose components are obtained by an analogous change of arrangement or joining the terms in brackets in each of the components ak + bk + …+pk. These components are numbers, they depend neither on the arrangement of components nor on the manner of joining them in brackets.