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Sachin Kaushal, Lovely Professional University Unit 10: Linear Functionals

Unit 10: Linear Functionals Notes

CONTENTS
Objectives
Introduction

10.1 Linear Functionals
10.2 System of Linear Equations
10.3 Summary

10.4 Keywords
10.5 Review Questions
10.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand in a better way the discussion of subspaces, linear equations and co-ordinates.
 See that a few examples of linear functional cited in this unit.
 Know the concept of dual basic vectors for the dual vector space V*.

 See that how to obtain the basis of the dual spaces which is done by examples.
Introduction

The concept of linear function is important in the study of finite dimensional spaces because the
linear functional method helps to organize and clarify the discussion of subspaces.
The method is illustrated by means of a few theorems and a few solved examples.

10.1 Linear Functionals

If V is a vector space over the field F, a linear transformation f from V into the scalar field F is also
called a linear functional on V. If we start from scratch, this means that f is a function from V into
F such that
f .(c ) cf ( ) f ( )

for all vectors and in V and all scalars c in F. The concept of linear functional is important in
the study of finite-dimensional spaces because it helps to organize and clarify the discussion of
subspaces, linear equations, and coordinates.

Example 1: Let F be a field and let a , ..., a be scalars in F. Define a function f on F by
n
1 n
( ,...,x
f x 1 n ) a x ... a x
1 1
n n
Then f is a linear functional on F . It is the linear functional which is represented by the matrix
n
n
[a ... a ] relative to the standard ordered basis for F and the basis {1} for F:
1 n
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