The Vector Space of all ordered n-tuples over a field F

The Vector Space of all ordered n-tuples over a field F.

Solution:

Let F be a field. An ordered set α = of n element of F is called an n- tuple over F. Let V be the totality of all ordered n-tuples over F i.e., Let

V = .

Now we shall give a vector space structure to V over the field F. For this we define equality of two n-tuples, addition of two n-tupes and multiplication of an n-tuple by a scalar as follows:

Equality of two n-tuples. Two element

α =

and  β =

of V are said to be equal if and only if for each i = 1, 2, .  .  ., n.

Addition composition in V. We define

α + β =

∀α = , β = ∈ V

Since are all elements of F, therefore α + β ∈ and thus V is closed with respect to addition on n-tuples.

Scalar multiplication in V over F. We define

aα = ∀a∈ F, ∀ α = ∈ V.

Since are all elements of F, therefore aα ∈ and thus V is closed with respect to scalar multiplication.

Now we shall see that V is a vector space for theses two composition.

Associative of addition in V. We have

=

= 

= 

=

=

Commutativity of addition in V. We have

Existence of additive inverse of each element of V. If

also if

∴ (0, 0, . . . 0) is the additive identity in V.

Existence of additive inverse of each element of V. If

then .

Also we have

= (0, 0, .  .  ., 0)

∴ is the additive inverse of .

Thus V is an abelian group with respect to addition. further we observe that

1. If a ∈ F and α = , β = , then

a(α + β) =

= aα + aβ

2. If a, b ∈ F and α = , then

(a + b)α =

= aα + bβ.

3. If a, b ∈ F and α = , then

(ab)α =

.

4. If 1 is the unity element of F and  α = , then

1α =

= α,

Hence V is a vector space over F. The vector space of all ordered n-tuples over F will be denoted by . Sometimes we also denote it by or . Here the zero vector i.e., 0 is the n-tuple (0, 0, .  .  ., 0)

Question:1: A field K can be  regarded as a vector space over any subfield F of K

Definition: ⇒Vector Space and Property of Vector Space