箭头空间：内积空间和Affine Geometry的一种方法
Arrow Spaces: An Approach to Inner Product Spaces and Affine Geometry
摘要：
给定一个假设的点集，对“箭空间”提出了一个代数系统公理。箭头被定义为两个点<T，H>的有序集合，分别命名为Tail和Head。这组箭头是一个箭头空间。箭头空间被公理化地赋予箭头空间“预内积“，其类似于欧几里得向量空间的内积。使用此箭头空间预内积，推导出箭头空间的各种性质，并与欧几里得向量空间的性质进行对比。向量空间的公理和它的相关内积被导出为遵循箭头空间的公理的定理，因为向量被严格地证明是箭头的等价类。应用箭头空间解决仿射几何中的几何问题。
英文摘要：
Given a postulated set of points, an algebraic system of axioms is proposed for an "arrow space'". An arrow is defined to be an ordered set of two points <T, H>, named respectively Tail and Head. The set of arrows is an arrow space. The arrow space is axiomatically endowed with an arrow space "pre-inner product" which is analogous to the inner product of a Euclidean vector space. Using this arrow space pre-inner product, various properties of the arrow space are derived and contrasted with the properties of a Euclidean vector space. The axioms of a vector space and its associated inner product are derived as theorems that follow from the axioms of an arrow space since vectors are rigorously shown to be equivalence classes of arrows. Applications of using an arrow space to solve geometric problems in affine geometry are provided.
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