Chapter List

Introduction

Starting from intuitively geometric objects, we treat

1. a point as a zero vector,

2. a directed segment (along a line) as a vector, and

3. two directed segments along the same or parallel lines as the same vector if both have the same length and direction.

And hence, we define two vector operations: scalar multiplication ? and addition and develop their operational properties. In the process, we single out the linear combination, dependence and independence among vectors as the main tools and establish the affine structures on a line, a plane and a space, respectively. Then, we extract the essence of concepts obtained and formulate, via rough ideas of linear isomorphism, the abstract sets ? ¹, ? ² and ? ³ as the standard one-dimensional, two-dimensional and three-dimensional vector spaces over the real field, respectively. So far, changes of coordinates in the same space are the most prominent results among all, which indicates implicitly the concepts of affine and linear transformations.

Then, we focus our attention to these mappings between spaces that preserve the ratios of signed lengths of segments along the same or parallel lines. They are affine transformations (see Secs. 1.4, 2.7 and 2.8), in particular, linear transformations if they map...