INTRODUCTION OF VECTOR
SPACES
DR. PRASANTA SINHA
DEPARTMENT OF MATHEMATICS
For B.Sc Part I
• The study of vectors relative to addition and multiplication by
real numbers can be generalized in two directions. Firstly it is
not necessary for us to restrict our study to ordered pairs or
ordered triples, instead one may take ordered triples.
Secondly we do not need confine ourselves to co-ordinates
being real numbers. The co-ordinates may be taken from any
field or more generally from any division ring.
• In Analytical Geometry and Mechanics, one come across the
concept of a vector as a directed magnitude, we do not need
not recall that definition here. From algebraic point of view
the vectors so defined have following salient features.
• In three dimensional Euclidean space, a vector v is
determined uniquely by its three components (ξ, η, λ )
(relative to a definite coordinate system) and conversely given
any order triple (ξ, η, λ ) of real numbers there exists a vector
having ξ, η, λ as its coordinates. Similarly in the two
dimensional Euclidean space, a vector has two components
ad is determined uniquely by its co-ordinates. Futher, the real
numbers are called scalars.
3
• There are three fundamental operations , namely addition of
vectors, the multiplications of a vector by a scalars and the
scalar product of vectors. Thus if V = (α, β, γ ) and V/ = (α/,
β/, γ/ ) are two vectors in R3, the three dimensional Euclidean
space, c is any scalar, then we define V + V/ = (α + α/, β + β/ , γ
+ γ/) and cV = (cα, cβ, cγ ) and scalar product (V, V/ ) = (αα/,
ββ/, γ γ/ ). First two operations on vectors play very significant
part in geometry, we endeavour to give a treatment of
vectors in a more general settings.
DEFINITION
• Definition 1.1 A system (V,F, +, .) where V is a non void set, F is a
field, + is a binary composition on V and . Is a mapping of F x V into
which for each element v ϵ V, α ϵ F determines an element α.v ϵ V
is called a vector space over a field F if it satisfies the following
conditionsV-1) (V, +) is an abelian group.
• V-2) For all x,y ϵ V, α,β ϵ F we have
• (i) α.(x+y) = α.x + α.y
• (ii) (α + β).x = α.x+β.x
• (iii) (αβ).x = α.(β.x)
• (iv) 1.x = x
• Every element of V is called a vector, every member of F is called a
scalar.
•
Example 1. For any field F, let V = {(α1, α2) : αi ϵ F }. Then V is a vector space over a
field F under vector addition and multiplication of a vector by a scalar given by
(α1, α2) + (β 1, β2) =( α1 + β 1, α2 + β2)
α (α1, α2) =( α α1, α α2).
Example 2. Let V be the additive group of polynomials in one variable over a field F.
For any α ϵ F and f(x) = α0 +α1x+ α2x2 +……………. + αnxn ϵ F[x]
define αf(x) = αα0 + αα1x+ αα2x2 +……………. + ααnxn
Then V becomes a vector space over F.
Example 3. Let V be a set of all continuous real-valued continuous function defined on
the closed interval [0,1]. For any f, g ϵ V and α ϵ R, define f +g and αf by
(f+g )(x) = f(x) +g(x)
(αf)(x) = αf(x) for all x in [0,1]
Under these operations, V becomes a vector space over R.
Example 4. For any field F and any positive integer n, the set of all n-tuples
(α1, α2, ………. , αn) of elementsof F, is avector space over F under addition and
multiplication of a vector by a scalar given by
(α1, α2, ………. , αn) + (β 1, β2,…………., β n) =( α1 + β 1, α2 + β2,…………., αn + β n)
α (α1, α2, ………. , αn) =( α α1, α α2, …………. , α αn).
This vector space is usually denoted by F(n) or by Fn.
• NOTATION 1.1 : VF will denote the vector space VF
over the field F.
Remark 1.For the sake of convenience, in future, we shall write αv in
place of α.v.
Remark 1.2. We give below an example to show that the condition V-2
(iv) viz. 1.x =x ∀ x ϵ V is independent of the remaining conditions V-1
and V-2 (i), ()ii , (iii).
Example 1.5. Consider V = {(α, β, γ) : α, β, γ ϵ R} Define
• (α, β, γ) + (α/, β/, γ/) = (α + α/ , β+ β/, γ + γ/)
•
λ (α, β, γ) = (λα, λβ, 0)
∀ α, β, γ, α/, β/, γ/ , λ ϵ R, It can be
easily verified that all the condition V-1, V-2 (i), (ii), and (iii) are
satisfied Now (3,2,6) ϵ V but 1(3,2,6) = (3,2,0) ≠ (3,2,6). Hence V is not
a vector space over R.
•
Lemma1. 6. Let V be a vector space over a field F and let 0V
and 0F be additive identities of V and F respectively, the the
following hold.
r 0V = 0V ∀ r ϵ F
0F v = 0V ∀ v ϵ V
–v = (-1) v ∀ v ϵ V
(-α)v = α (-v) = - ( αv)
If αv = 0V then either α = 0F or v = 0V.
NOTE 1.1. In future we shall use the same symbol 0 and O for
the zero vector 0V and for the zero scalar 0F, for any vector vF.
it should be clear to the readers from the context for which
thing 0 or O is being used at a particular occasion.
SUBSPACES
• Definition 1.4 a non void sub set W of a vector
space of VF is said to be a subspace of VF if W is a
vector space itself over the same field . The
equivalent description of a subspace is given
below
• S-1. For any a,b ϵ W a+b ϵ W
• S-2. For any a ϵ W and r ϵ F, ra ϵ W.
• It is immediate from the definition that (0)and V
are both subspaces of V these are trivial subspace
of V. Any subspace W of VF which is different
from(0) and V is called a proper subspace of V.
Lemma 1.2. If W is a subspace of a vector space VF, then W is a
subgroup of (V, +) and W is vector space over F under induced addition
and multiplication of vectors by scalars.
Proof. For any x, y ϵ W, we have (-1)y ϵ W by S-2; so by S-1, x + (-1)y ϵ W
→ x – y ϵ W. This proves that W is a subgroup of (V,+).
The property V-2 in definition 1.1 is satisfied by elements of W, since W
⊆ V and the same property holds for V and also the composition in W
are all induced by those in V. Hence W is a vector space over F under
induced addition and the multiplication of vector by scalars.
Examples
• Example1. 6. Consider R3 = {(a1, a2, a3) : ai ϵ R}
Let W1 = {(0, a2, a3) : ai ϵ R }
W2 = {(a1, a2, a3) : ai ϵ R and a1 + a2, + a3 =0}. It can be
easily seen that W1 and W2 both satisfy the
condition which define a subspace. Consequently
W1 and W2 are both subspaces of R3
• Example 1.7. Let C be the set of all complex
numbers, C is a vector space over R, the field of
real numbers. W = {ib : b ϵ R , i= √ (-1)} is a
subspace of C.
Theorem 1.1. A non void subset W of a vector space VF is
a subspace of VF iff it satisfies the following condition:
a, b ϵ F; x, y ϵ W implies ax +by ϵ W.
Proof. It is immediate.
Theorem 1.2. Intersection of any family of subspaces of
a vector space
is a subspace.
Proof. It is quite obvious.
Question: Does the above result hold good for finite
union of subspaces?
Answer: Unfortunately the answer is in negation. in
fact, it does not hold for union any two subspaces,
which follows from the following example.
Example 1.8. Consider R2 is a vector space over the
field of real numbers R and two subspaces
W1 = {(a,0) : a ϵ R }
W2 = {(0,b) : b ϵ R }
are two subspaces of R2. Let W = W1 U W2.
Now (1,0) ϵ W1 , (0,1) ϵ W2, but (1,1) is neither in W1
W2. So, that, (1,1)= (1,0) +(0,1) ∉ W = W1 U W2.
Consequently, W1W2 is not a subspace of R2.
Definition 1.5. Let X be subset of a vector space V,
then a subspace W of V is said
to be spanned by or generated by X if (i) X ⊆ W
(ii)For any subspace W/ of V ,
X ⊆ W/ implies W ⊆ W/ . i.e., smallest subspace
which generates X.
Notation 1.2 . <X> will denote the subspace
spanned by X.
Further if X consists
of elements x1, x2,…. xn, then we simply write
<X> = < x1, x2,…. xn>.
Theorem 1.3. For any nonvoid subset X = {x1, x2,…. xn}
the subspace spanned by or generated by X is the set of
all of all vectors of the form a1x1 + a2x2 +…….+ anxn,
where ai ϵ F, for i=1,2,…,n.
Proof.
Definition 1.5. For any finite number of vectors x1, x2,….
xn in a vector space V and the scalar a1, a2,…., an in F, the
vector a1x1 + a2x2 +…….+ anxn is called a linear
combination of x1, x2,…. xn.
Definition 1.6. For any two subspaces W1 and W2 of a
vector space V, their sum W1 + W2 is the set given by
{w1 + w2 : w1 ϵ W1 , w2 ϵ W2}
Theorem 1.3. For any two subspaces W1 and W2 of a vector space
V , W1 + W2 is a subspace of V spanned by W1 U W2.
Proof. Firstly we show that W1 + W2 is a subspace of V. Clearly 0
= 0+0 ϵ W1 + W2 So, W1 + W2 ≠ φ . Let x, y ϵ W1 + W2 , a, b ϵ F .
Then x = x = w1 +w2, y = w3 +w4, w1, w3 ϵ W1 , w2, w4 ϵ W2.
Thus ax + by = (a w1 + b w3) + (a w2 + b w4) ϵ W1 + W2.
Hence, W1 + W2 is a subspace of V.
Now we show that W1 + W2 is spanned by W1U W2. For any
w1 ϵ W1 as 0 ϵ W2 we have w1 = w1 + o ϵ W1 + W2 so, we have
W1 ⊆ W1 + W2, Similarly, we can show that W2 ⊆ W1 + W2, So,
W1 U W2 ⊆ W1 + W2. Let W be any subspace of V containing W1
U W2. Then given w1 ϵ W1, w2 ϵ W2, we have w1, w2 ϵ W1 U W2 ⇛
w1, w2 ϵ W ⇛ w1 +w2 ϵ W, since V is closed under vector addition,
it implies that W1 + W2 ⊆ W.
Example 1.9. For any field, consider the vector space F(3) . e1 = (1,0,
0), e2 = (0,1, 0), e3 = (0,0, 1), are in (1,0, 0),.
Then x = a1(1,0, 0) + a2(0,1, 0) + a3(0,0, 1) = a1e1 + a2 e2+ a3e3. Hence
every member of F(3) is a linear combination of e1, e2, e3.
Consequently, { e1, e2, e3} spans F(3). In general if we consider F(n) and
define e1 = (1,0,….., 0), e2 = (0,1, 0,….,0), ……….en = (0,0,…, 1), then
F(n) is spanned by the n vectors e1, e2,……, en.
Example 1.10. Consider R(3) , x1 = (1,1,0), x2 = (1,2,0) ϵ R3. Let W be
the subspace spanned by
{ x1 , x2}. Then x1 , x2 ϵ W ⇛ x2 – x1 ϵ W ⇛ (0,1,0) ϵ W ⇛ x1 – (0.1.0) ϵ
W. Thus, e1 = (1,0, 0), e2 = (0,1, 0) ϵ W ⇛ for every a, b ϵ R,
a e1 + b e2 = (a,b,0) ϵ W.
Let W/ = {(a,b,0): a,b ϵ R}. W/ is a subspace of R3 and as seen above
W/ ⊆ W.
. Now x1 = e1 + e2 ϵ W/, x2 = e1 + 2e2 ϵ W/ since e1 , e2 ϵ W/. Thus the
subspace W = < x1 , x2> ⊆ W/. Hence W = W/ .
Linear Independence
• Definition 1.6. Let x1, x2,…. xn be a finite number of members
(not necessarily all distinct) of avector space V. these
vectors are said to be linearly dependent (L.D) if for some
a1, a2,……, an ϵ F with at least one of them non zero, a1 x1 +
a2 x2 +……+ an xn = 0. These vectors are said to be linearly
Independent (L.I) if for all ai ϵ F (I = 1,2,….,n), a1 x1 + a2 x2
+……+ an xn = 0 ⇛ ai =0 for all i.
• Example 1.8. Consider V = {(a,b) : a,b ϵ F} where F is a field.
The vectors e1 = (1, 0), e2 = (0,1) in V are L.I, since for any
r,s ϵ F, r e1 + s e2 = 0 ⇛ (r,0) + (0,s) = (0,0) ⇛ r = 0 , s = 0.
Further notice that any vector (a, b) ϵ V can be expressed as
ae1 + be2 as a linear combination of e1 and e2.
Theorem 1.4. If v1, v2,……….. vn are L.I. in a vector
space V, then each element of the subspace W
spanned by them is expressible uniquely as a linear
combination of v1, v2,……….. vn.
Theorem 1.5. Let u1, u2,………, un be any n linearly
independent vectors in a vector space V. Any n+1
vectors v1, v2,……….. vn, vn+1, each of which is a linear
combination of u1, u2,………, un are L.D.
Corollary 1.6. If v1, v2,……….. vn is a L.I. subset of a
vector space V, then any subset W of V having more
than n vectors each of which can be expressed as a
linear combination of v1, v2,……….. vn must be linearly
dependent.
BASE
• We conclude section with the concept of Base.
• Definition 1.7. A subset B of a vector space V is called a
basis of V if (i) B is linearly independent and (ii) B spans V.
• Definition 1.8. A vector space V is finitely generated if it
has a finite subset which spans V itself.
• Example 1.9. For any field F, consider F(n), the vector space
consisting of all n-tuples of the type (α1, α2,. ……, αn) over F.
Let e1 = (1,0,0,……..,0), e2 = (0,1,0,……..,0),……..en =
(0,0,0,……..,1). Then it can be easily seen that e1, e2,……., en
are L.I. and they span F(n). Thus { e1, e2,……., en } is a basis of
F(n) and F(n) is finitely generated.