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Richa Nandra, Lovely Professional University Unit 6: Computation Concerning Subspaces

Unit 6: Computation Concerning Subspaces Notes

CONTENTS
Objectives
Introduction

6.1 Computation Concerning Subspaces
6.2 Illustrative Examples
6.3 Summary

6.4 Keywords
6.5 Review Questions
6.6 Further Readings

Objectives

After studying this unit, you will be able to:

 See that the units (3), (4) and (5) are quite suitable to find if a set of vectors ,  ,... are
1 2 m
linearly independent.
 Determine whether another vector is a linear combination of  ,... .
1 m
 See that the detailed examples in this unit clarify most ideas covered in the last few units.

Introduction

This unit mostly summarizes the ideas of row-operations in helping to find out the basis of a
vector-subspace.

One can understand how a vector  belongs to the vector sub-space spanned by the basis vectors.

6.1 Computation Concerning Subspaces

In this unit we should like to show how elementary row operations helps us in understanding
in a concrete way the subspaces of F . This discussion applies to any n-dimensional vector space
n
over the field F, if one selects a fixed ordered basis  and describes each vector  in V by the
n-tuple (x , x ,...,x ) which gives the co-ordinates of  in the ordered basis .
1 2 n
Suppose we are given m vectors  ,..., in F . We consider the following questions.
n
1 m
1. How does one determine if the vectors  ,  ,..., are linearly independent? How does
1 2 m
one find the dimension of the subspace W spanned by these vectors?
n
2. Given  in F , how does one determine whether  is a linear combination of  ,..., , i.e.,
1 m
whether  is in the subspace W?
3. How can one give an explicit description of the subspace W?
The third question is a little vague, since it does not specify what is meant by an ‘explicit
description’; however, we shall clear up this point by giving the sort of description we have in
mind. With this description, questions (1) and (2) can be answered immediately.

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