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Home / Linear Algebra / Vector Spaces| Chapter 4|linear algebra

Vector Spaces| Chapter 4|linear algebra

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:

[A1]   (u+v)+w=u+(v+w)

[A2]   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

[A3]   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

[A4]   u+v=v+u

[M1]   k(u+v) = ku+kv, for any scalar k∊K

[M2]   (a+b)u = au+bu, for any a,b∊K

[M3]    (ab)u=a(bu),   for any a,b∊K

[M4]   1u=u, for the unit scalar 1∊K

 

Space Kn

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

(i) Vector addition: (a1, a2, ————- an)+ (b1, b2, ————- bn)

=( a1+ b1, a2+ b2, ————————-, an+ bn)

(ii) Scalar multiplication: k(a1, a2, ————- an) =(ka1, ka2, ———,kan)

Polynomial space P(t)

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

P(t)=a0+a1t+ a2t2+——————+ asts       (s=1, 2, —————-)

Where the coefficients ai 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 Pn(t)

Let Pn(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)=a0+a1t+ a2t2+——————+ asts

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.

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