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Sachin Kaushal, Lovely Professional University Unit 1: Vector Space over Fields

Unit 1: Vector Space over Fields Notes

CONTENTS
Objectives
Introduction

1.1 Sets
1.2 Groups
1.3 Rings

1.4 Fields
1.5 Vector Spaces
1.6 Summary
1.7 Keywords
1.8 Review Questions

1.9 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the concept of abstract sets
 Explain the concept of functions
 Discuss the abstract groups and their properties
 State the properties of rings and fields
 Understand abstract vector space. This will help you to understand sub-spaces, bases and
dimension in the next units
 Know that this unit is a prerequisite to understand the next few units.

Introduction

In this unit the idea of set theory is explained. The unit also deals with functions and mapping.

The ideas of rings and fields help us to study vector spaces and their structure. This unit briefly
explains the properties of vector spaces which are useful in understanding the vector sub-spaces,
bases dimensions and co-ordinates.
1.1 Sets

The concept of set is fundamental in all branches of mathematics. A set according to the German
mathematician George Cantor, is a collection of definite well-defined objects of perception or thought.
By a well defined collection we mean that there exists a rule with the help of which it is possible
to tell whether a given object belongs or does not belong to the given collection. The objects in
sets may be anything: numbers, people, animals etc. The objects constituting the set are called
elements or members of the set.

LOVELY PROFESSIONAL UNIVERSITY 1