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Sachin Kaushal, Lovely Professional University Unit 4: Co-ordinates

Unit 4: Co-ordinates Notes

CONTENTS
Objectives

Introduction
4.1 Co-ordinates
4.2 Change of Basis from One Ordered Basis to Another

4.3 Summary
4.4 Keywords
4.5 Review Questions
4.6 Further Readings

Objectives

After studying this unit, you will be able to:

 See that the dimension and basis of a vector space V over the field F help us in introducing
the co-ordinates of a vector.
 Understand how to go from one basis to another basis with the help of an invertible
matrix.
 See that the solved examples help you to find the invertible matrix and hence the
co-ordinates of the vector in the new basis can be found out.

Introduction

For an abstract vector space V over the field F can be spanned by a set of independent vectors
which form the basis of the vector space V.

There are more than one way of finding the basis and so it is important to know the relation
between one basis over the other.

4.1 Co-ordinates

So far we have dealt with basis and dimension in the unit 3. We also showed the linear
independence and dependence of vectors. The dimension of a vector space is the number of basis
vectors of the vector space V over the field. The standard basis for a three dimensional vector
space is taken as

l 1 1,0,0

l 0,1,0
2
l 0,0,1
3
and they form an independent set of vectors and span the whole V over the field R.
3

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