# Chapter 4

# CONCEPTS FROM GEOMETRY

# 4.1 Line Segments

In the following analysis we concern ourselves only with ^{n}. The elements of this space are the *n*-component vectors ** x** = [

*x*

_{1},

*x*

_{2},…,

*x*

_{n}]

^{}.

The *line segment* between two points ** x** and

**in**

*y*^{n}is the set of points on the straight line joining points

**and**

*x***(see Figure 4.1). Note that if**

*y***lies on the line segment between**

*z***and**

*x***, then**

*y*where α is a real number from the interval [0,1]. The equation above can be rewritten as ** z** = α

**+ (1 − α)**

*x***. Hence, the line segment between**

*y***and**

*x***can be represented as**

*y*# 4.2 Hyperplanes and Linear Varieties

Let *u*_{1}, *u*_{2},…, *u*_{n}, *v* , where at least one of the *u*_{i} is nonzero. The set of all points ...

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