Summary Notes

Graduate Course



B.A. (PROGRAMME) 1 YEAR
ALGEBRA AND CALCULUS

SM – 1 PART – A: (ALTEBRA)
VECTOR SPACES AND MATRICES




CONTENTS

Lesson 1 : Vector Spaces
Lesson 2 : Matrices : Basic Concepts
Lesson 3 : Elementary Operations on a Matrix and Inverse of a Matrix
Lesson 4 : Rank of a Matrix
Lesson 5 : Systems of Linear Equations
Lesson 6 : The Characteristic Equation of a Matrix




Editor:
Dr. S.K. Verma




SCHOOL OF OPEN LEARNING
UNIVERSITY OF DELHI
5, Cavalry Lane, Delhi-110007

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,Session 2012-2013 (1500 Copies)




© School of Open Learning




Published By: Executive Director, School of Open Learning, 5, Cavalary Lane, Delhi-110007
Laser Composing By : M/s Computek System (2012-13)


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, LESSON 1


VECTOR SPACES


1.1 Introduction
You are already familier with several algebric structures such as groups, rings, integral domains
and fields. In this lesson we shall tell you about another equally important algebric structures, namely,
a vector space.
Let V be a non-emply set and let F be a field. Let us agree to call elements of V vectors and
elements of F scalars.
A maping from V × V to V will be called addition in V and a mapping from F × V to V will
be called multiplication by a scalar multiplication, V is said to a vector space over F if addition and
scalar multiplication satisfy certain properties. Of course, these conditions are to be chosen in such a
manner that the resulting algebric structure is rich enough to be useful. Before presenting the definition
of a vector space, let us note that addition in V is denoted by the symbol ‘+’, and scalar multiplication
is denoted by juxtaposition, i.e., if x ∈ V, y ∈ V, and α ∈ F, the x + y denoted the sum of x and
y, and αx denotes the scalar multiople of x by α.
Defintion 1. A non-empty set V is said to be a vector space over a field F with respect to addition
and scalar multiplication if the following properties hold.
V 1 Addition in V is associative, i.e.,
x + (y + z) = (x + y) + z, for all x, y, z, ∈ V
V 2 There exists of natural element for additon in V, i.e., there exists an element 0 Î V such
that
x + 0 = 0 + x = x, for all x ∈ V
V 3 Every element of V possesses a negaive (or addtion inverse), i.e., for each x ∈ V, there
exists an element y ∈ V such that
x + y = y + x = 0.
V 4 Addition in V is commulative, i.e., for all elements x, y ∈ V,
x+ y = y+ x
V 5 Associtiavity of scalar multiplication, i.e.,
i.e., α(βx) = (α β) x, for all α, β, ∈ F and x ∈ V
V 6 Property of 1. For all x∈V,
1x = x, where 1 is the multiplicative identity of F.
V 7 Distributivity properties for all α, β ∈ F and x, y ∈ V.
(α + β)x = αx + βx
α(x + y) = αx + αy
Remarks 1. The first of the two distributivity properties stated in V 7 above is generally called
distributivity of scalar multiplication over addition in F, and the second of the two distributibity properties
is called distributivity of scalar multiplication over addition in V.


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, 2. We generally refer to properties V 1 – V 7 above by saying that (V, +) is a vector space over
F. If the underlying field F is fixed, we simply say that (V, +,) is a vector space, and do not make
an explicit reference to F.
In case, the two vector space compositions are known, we denote a vectors space over a field’
F by the symbol V(F). If there is no chance of confusion about the underlying field, then we simply
talk of ‘the vector space V’.
3. You might have observed that the axions V 1 to V 4 simply assert the V is an abelian group
for the composition ‘+’. In view of the we can re-state the definition of a vector space as follows:

2. Definnition and Exampls of a Vector Space
Defintion 2. A triple (V, +,) is said to be a vector space over a field F if (V, +) is an abelian
group, and the following properties are satisfied :
α(βx) = (αβ)x, ∀α, β ∈ F and ∀ x ∈ V
1x = x, ∀x, β ∈ V, where 1 is the multiplicative identity of F
(α + β)x = αx + βx, ∀α, β ∈ F, and ∀x, y ∈ V
α (x + y) = αx + αy, ∀α ∈ F and ∀ x, y ∈ V
We shall now consider some examples of vector spaces.
Example 1. Let R be the set of number (R, +) is vector space over R. The addition is addition
in R and scalar multiplication is simply multiplication of real numbers.
It is easy to verify that all the vector space axioms are verified. In fact, V1-4 are satisfied because
R is an abelian group with respect to addition, V5 is nothing but the associative property of multipication,
V6 is the property of the multiplicative identity in (R, +,) and the properties listed in V7 are nothing
but the distributivity of multiplication over addition.
Example 2. (C, +, ) is a vector space over C
Example 3. (Q, +, ) is a vector space over Q.
Example 4. Let F be any field. F is a vector space over itself for the usual compositions of addition
and multipication (to be called scalar multiplication) in F.
Example 5. C is a vector space over R, and R is a vector space over Q.
Example 6. R is not a vector space over C. Observe that if α ∈ C and x ∈ R, the αx is not
in R. Therefore the multiplication composition in R fails to give rise to the scalar multiplication composition.
The examples considered above are in a way re-labelling of the field properties C, R or Q. We
shall now consider some examples of a different type.
Example 7. Let V be the set of all vectors in a plane. You know that addition of two vectors
is a vector, and that V is a group with respect to sum of vectors. Let us take addition of vectors as
the first compostion for the purpose of our example. Also, we know that if d be any vector and k be
any real numbers, then k d is a vector. Let us take R as the underlying field and multiplication of
vector by a scalar as the second vector space composition. It is easy to see that V is vector space over
R for these two compositons.
Example 8. Let R3 be the set
{( x1 , x2 , x3 ) : x1 , x2, x 3 ∈ R}



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