Vector Space and properties of Vector space

Vector Space

Defnition: Let (F, +, .) be a field. The element of F will be called scalars. Let V be a non-empty set whose elements will be called Vectors. Then V is vector space over the field. If

1.  There is defined an internal composition in V called addition of vectors and denoted by ‘+’. Also for the composition V is an abelian group.

2. There is an external composition in V over F called scalar multiplication and denoted multiplicatively i.e. aα ∈ V. In other words V is closed with respect to scalar multiplication.

3. The two compositions i.e. scalar multiplication and addition of vectors satisfy the following postulates:

(i) a (α + β) = aα + bβ  for all a ∈ F and for all α, β ∈ V.

(ii) (a + b) α = aα + bα for all a, b ∈ F and for all α ∈ V

(iii) (ab)α = a(bα) for all a, b ∈ F and for all α ∈ V.

(iv) Iα = α for all α ∈ V and I is the unity element of the field F.

When V is a vector space over the field F, we shall say that V(F) is a vector space. If F is the field R of real numbers, V is called real vector space.

Some example of Vector space

(i) V = R^n = \{a_1, a_2, a_3 .  .  ., a_n\}:a_i \in R

F = R (Real number)

Vector space = R^n(R)

(ii) V = P(x) = \{a_0+a_1x+a_2x^2  + .  .  . + a_nx^n|a_i \in R\}

F = R

V(F) is a Vector Space.

(iii) V =\{f|f:R \rightarrow R\} f is real valued function.

F = R

V(R) is a Vector Space

(iv) Every field is Vector Space over its Subfield.

Note 1:

For V to be an abelian group with respect to addition of vectors, we must have the following condition satisfied:

(i) α + β ∈ V for all α, β ∈ V.

(ii) α + β = β + α for all α, β ∈ V.

(iii) α +(β + γ) =( α +β) + γ  for all α, β, γ ∈ V.

(iv) There exists an element 0 ∈ V such that 0 + α ∈ V for all  α ∈ V.

The element 0 ∈ V will be called the zero vector. It is additive identity  in V

(v) To every vector α ∈ V there exists a vector -α ∈ V such that -α + α = 0. Thus each vector should possess aditive inverse . The vector -α is called the negative of the vector α.

Note 2:

There should also be no confusion about the use of the word vectors. Here by vector we do not mean the vector quantity which we have defined in vector algebra. Here we shall call the elements of the set V as Vectors.

Note 3:

In vector space we shall be dealing with two types of zero elements. One is the zero vector and the other is the zero element of the field F i.e. the 0 scalar. The student may use the same symbol 0 to denote the zero vector as well as the zero scalar.

Note 4:

We shall usaally use the lower case Greek letter  α, β, γ etc for vector elements of V and the lower case Latin letters a, b,c etc for scalars elements of the field F.

General Properties of Vector Space

 Theorem 1.

Let V(F) be a vector space and 0 be the zero vector of V. Then

(i)  a0 = 0  for all a∈ F, 0 ∈ V.

(ii)  0α =for all 0,α ∈ V, 0 ∈ F.

(iii) a(-α) = -(aα) for all a∈ F, α ∈ V.

(iv) (-a)α = -(aα) for all a∈ F, α ∈ V.

(v) a(α – β) = aα – aβ for all  a∈ F, α, β ∈ V.

(vi)  aα =⇒ a = 0 or α = 0

Proof:

(i) We have , a0 = a(0 + 0)

⇒  a0 = a0 + a0

⇒  0 +a0 = a0 + a0           [Since, a0∈ V and 0 +a0  = a0 ]

Therefore from right cancellation law in V, we get

0  = a0

(ii) We have, 0α = (0 + 0)α

⇒  0α = 0α  + 0α

0 + 0α = 0α  + 0α       [Since,0α V and 0 + 0α  = 0α ]

Right cancellation law in V, we get

0α = 0

(iii) we have,   a[α + (-α)] = aα + a(-α)

⇒ a0 =  aα + a(-α)

⇒   0 = aα + a(-α)     [Since, a0 = 0]

⇒ a(-α) = – (aα) .

(iv) We have, [a + (-a)]α = aα + (-a)α

⇒ 0α = aα + (-a)α

⇒  0 = aα + (-a)α

⇒  (-a)α  = -(aα).

(v) We have, a(α – β) = a[α + (-β)]

⇒  a(α – β) = aα +a(-β)

⇒  a(α – β) = aα – aβ          [Since, a(-β) = -(aβ)]

(vi) Let aα = 0 and a ≠ 0. then a^{-1} exist because aa is non- zero element of the field F.

∴     aα = 0 a^{-1}(a\alpha) = a^{-1}0

\Rightarrow (a^{-1} a)α =0

⇒ Iα =⇒ α = 0

Again

Let aα = 0 and α ≠ 0. then α^{-1} exist because aa is non- zero element of the field F.

∴     aα = 0 (a\alpha)\alpha^{-1} = 0\alpha^{-1}

\Rightarrow a(\alpha^{-1}\alpha) =0

⇒ aI = 0 ⇒ a = 0

Theorem 2:

Let V(F) be vector space. Then

(i) If a, b ∈ F and α is non zero element of V, we have aα = bα ⇒ a = b.

(ii) If α, β ∈ V and a is non- zero elemeent of F, we have aα = aβ ⇒ α = β.

Prrof:

(i) We have, aα = bα

⇒ aα – bα = 0

⇒  (a – b)α = 0

But α ≠ 0. Therefore  (a – b)α = 0

⇒ a – b = 0

⇒  a = b.

(ii) We have, aα = aβ ⇒ aα – aβ  =

⇒ a(α – β) =

But a ≠ 0. Therefore  a(α – β) = 0

⇒  α – β  = ⇒  α  =  β .

Vector Subspaces

Definition: Let V be a vector space over the field F and W ⊆ V. Then W is called a subspace of V if W itself is a vector space over F with respect to the operations of vector addition and scalar multiplication in V.

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