Elementary Linear Algebra, 11th Edition

CHAPTER 3Euclidean Vector Spaces

CHAPTER CONTENTS

1. 3.1 Vectors in 2-Space, 3-Space, and $n$-Space
2. 3.2 Norm, Dot Product, and Distance in $R^n$
3. 3.3 Orthogonality
4. 3.4 The Geometry of Linear Systems
5. 3.5 Cross Product

INTRODUCTION

Engineers and physicists distinguish between two types of physical quantities—scalars, which are quantities that can be described by a numerical value alone, and vectors, which are quantities that require both a number and a direction for their complete physical description. For example, temperature, length, and speed are scalars because they can be fully described by a number that tells “how much”—a temperature of 20°C, a length of 5 cm, or a speed of 75 km/h. In contrast, velocity and force are vectors because they require a number that tells “how much” and a direction that tells “which way”—say, a boat moving at 10 knots in a direction 45° northeast, or a force of 100 lb acting vertically. Although the notions of vectors and scalars that we will study in this text have their origins in physics and engineering, we will be more concerned with using them to build mathematical structures and then applying those structures to such diverse fields as genetics, computer science, economics, telecommunications, and environmental science.

3.1 Vectors in 2-Space, 3-Space, and n-Space

Linear algebra is primarily concerned with two types of mathematical objects, “matrices” ...