Chapter 3 - Vector spaces

Abstract

This chapter focuses on the vector spaces. A nonempty set F of scalars (real or complex numbers) on which the operations of addition and multiplication are defined, so that to every pair of scalars α, β ∊ F there corresponds scalars α+β and αβ in F is called a field. A vector space V over the field F is a set of vectors on which the operations of addition and multiplication by a scalar are defined, that is, for every pair of vectors X, Y ∊ V, there corresponds a vector X + Y ∊ V and also α X ∊ V, where α is a scalar. If F is the field R of real numbers, V is called a real vector space and if F is the field C of complex numbers, V is a complex vector space. A nonempty subset U of a vector space V is called a subspace of V or a linear manifold if it is closed under the operation of addition and scalar multiplication. A subspace U of a vector space V is itself a vector space and as all vector spaces, it always contains the null vector 0. The whole space V and the set consisting of the null vector alone are two special examples of subspaces.