Page 67 - DMTH502_LINEAR_ALGEBRA

Sachin Kaushal, Lovely Professional University Unit 2: Vector Subspaces

Unit 2: Vector Subspaces Notes

CONTENTS
Objectives

Introduction
2.1 Vector Subspace
2.2 Illustrative Examples

2.3 Summary
2.4 Keywords
2.5 Review Questions
2.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Understand the concept of a vector subspace

 Know more about subspaces by worked out examples
 See that a subspace has all the properties of a vector space.
Introduction

The unit one is the basis of the next five units. This unit is also based on the ideas of a vector
space.
The subspace idea will help us in understanding the concepts of basis and dimension as well as
how to set up the co-ordinates of a vector.

2.1 Vector Subspace

Let V be a vector space over a field F. Then a non-empty subset W of V is called a vector subspace
of V if under the operations of V, W itself, is a vector space of F. In other words, W is a subspace
of V whenever w w 2 W , , F w 1 w 2 W .
,
1
Algebra of Subspaces

Theorem 1: The intersection of any two subspaces w and w of a vector spaceV F is also a
1 2
subspace of V F .

Proof: w w is non-empty because at least o w and w both.
1 2 1 2
Let ,u v w w and , F
1 2
Then u w 1 w 2 u w 1 and u w 2

LOVELY PROFESSIONAL UNIVERSITY 61