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Sachin Kaushal, Lovely Professional University Unit 3: Bases and Dimensions of Vector Spaces

Unit 3: Bases and Dimensions of Vector Spaces Notes

CONTENTS
Objectives

Introduction
3.1 Linear Dependence and Linear Independence of Vectors
3.2 Basis and Dimension of a Vector Space

3.3 Summary
3.4 Keywords
3.5 Review Questions
3.6 Further Readings

Objectives

After studying this unit, you will be able to:

 See that in dealing with a finite dimensional vector space V over the F, we sometime
enquire whether a set of vectors is dependent or independent set.
 Understand that if you find a set of vectors as independent set in a vector space V then this
set of vectors forms the basis of the space V and the number of vectors in the sets defines
the dimension of the space V.

Introduction

In this unit we explain the concept of linear dependence and linear independence of the set of
vectors.
The number of independent set of vectors determines the dimension of the vector space and the
set of independent vectors forms the basis of the vector space.

3.1 Linear Dependence and Linear Independence of Vectors

Linear Dependence: Let V F be a vector space and let S u 1 ,u 2 ,...u n be a finite subset of V .
Then S is said to be linearly dependent if there exists scalars , ... , F not all zero, such that
1 2 n
u u ... u 0.
1 1 2 2 n n
Linear Independence: Let V F be a vector space and let S u ,u ,...u be finite subset of V.
1 2 n
Then S is said to be linearly independent if

n
a u i 0, 1 . F
u
i 1
holds only when 0, i 1,2,... .
n
i

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