Elementary Linear Algebra, 11th Edition

CHAPTER 4General Vector Spaces

CHAPTER CONTENTS

• 4.1 Real Vector Spaces
• 4.2 Subspaces
• 4.3 Linear Independence
• 4.4 Coordinates and Basis
• 4.5 Dimension
• 4.6 Change of Basis
• 4.7 Row Space, Column Space, and Null Space
• 4.8 Rank, Nullity, and the Fundamental Matrix Spaces
• 4.9 Basic Matrix Transformations in R² and R³
• 4.10 Properties of Matrix Transformations
• 4.11 Geometry of Matrix Operators on R²

INTRODUCTION

Recall that we began our study of vectors by viewing them as directed line segments (arrows). We then extended this idea by introducing rectangular coordinate systems, which enabled us to view vectors as ordered pairs and ordered triples of real numbers. As we developed properties of these vectors we noticed patterns in various formulas that enabled us to extend the notion of a vector to an $n$-tuple of real numbers. Although $n$-tuples took us outside the realm of our “visual experience,” it gave us a valuable tool for understanding and studying systems of linear equations. In this chapter we will extend the concept of a vector yet again by using the most important algebraic properties of vectors in $R^n$ as axioms. These axioms, if satisfied by a set of objects, will enable us to think of those objects as vectors.

4.1 Real Vector Spaces

In this section we will extend the concept of a vector by using the basic properties of vectors in $R^n$ as axioms, which if satisfied by a set of objects, guarantee that those objects behave like familiar vectors.

Vector Space Axioms

The following ...