**Definition:** Let V be a nonempty set of vectors with two operations:

(i) Vector addition: This assigns to any u,v ∊V, k∊K a sum u+v in V.

(ii) Scalar multiplication: This assigns to any u ∊V, k ∊K a product ku ∊V.

Then V is called a vector space if the following axioms hold for any vectors u,v,w ∊V:

[A_{1}] (u+v)+w=u+(v+w) [A

_{2}] There is a vector in V, denoted by 0(zero) and called the zero vector, such that for any u∊V,

u+o=0+u=u

[A_{3}] For each u∊V, there is a vector in V, denoted by –u, and called the negative of u, such that

u+(-u)=(-u)+u=0

[A_{4}] u+v=v+u [M

_{1}] k(u+v) = ku+kv, for any scalar k∊K [M

_{2}] (a+b)u = au+bu, for any a,b∊K [M

_{3}] (ab)u=a(bu), for any a,b∊K [M

_{4}] 1u=u, for the unit scalar 1∊K

**Space K ^{n}**

Let K be an arbitrary field. The notion K^{n} is frequently used to denote the set of all n-tuples of elements in K. Here K^{n} is a vector space over K using the following operations:

(i) Vector addition: (a_{1}, a_{2}, ————- a_{n})+ (b_{1}, b_{2}, ————- b_{n})

=( a_{1}+ b_{1}, a_{2}+ b_{2}, ————————-, a_{n}+ b_{n})

(ii) Scalar multiplication: k(a_{1}, a_{2}, ————- a_{n}) =(ka_{1}, ka_{2}, ———,ka_{n})

**Polynomial space P(t)**

Let P(t) denote the set of all polynomial real polynomials of the form

P(t)=a_{0}+a_{1}t+ a_{2}t^{2}+——————+ a_{s}t^{s} (s=1, 2, —————-)

Where the coefficients a_{i} belong to a field K. Then P(t) is a vector space over K using the following operations:

(i) Vector addition: Here p(t)+q(t) in P(t) is the usual operation of addition of polynomials.

(ii) Scalar multiplication: Here kp(t) in P(t) is the usual operation of the product of a scalar k and a polynomial p(t).

**Polynomial space P _{n}(t)**

Let P_{n}(t) denote the set of all polynomials p(t) over a field K, where the degree of p(t) is less than or equal to n, that is,

P(t)=a_{0}+a_{1}t+ a_{2}t^{2}+——————+ a_{s}t^{s}

Where s≤n.

**Function Space F(X)**

Let X be a non empty set and let K be an arbitrary field. Let F(X) denote the set of all functions of X into K. Then F(X) is a vector space over K with respect to the following operations:

(i) Vector addition: The sum of two functions f and g in F(X) is the function f+g in F(X) defined by

(f+g)(x) = f(x) + g(x) ∀x∊X

(ii) Scalar multiplication: The product of a scalar k∊K and a function f in F(X) is the function kf in F(X) defined by

(kf)(x) =kf(x) ∀x∊X

**Fields and subfields:**

Suppose a field E is an extension of a field K, that is, suppose E is a field K as a subfield. Then E may be viewed as a vector space over K using the following operations.

(i) Vector addition: Here u+v in E is the usual addition E.

(ii) Scalar multiplication: Here ku in E, where k ∊K and u∊E, is the usual product of k and u as element of E.