Question

define vector space​

Answer 1

Step-by-step explanation:

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Answer 2

Vector space : Let V be a non empty set with two operations :

1. Vector addition : This assigns to any u , v ∈ V , a sum u + v in V .
2. Scalar multiplication : This assign to any u ∈ V , k ∈ K , a product ku ∈ V .

Then , V is called a vector space (over the field F) if the following conditions hold :

1. Associative property : (u + v) + w = u + (v + w) ∀ u , v , w ∈ V .
2. Existence of identity : There is a vector in V , denoted by 0 and called the zero vector , such that u + 0 = 0 + u = u ∀ ∈ V .
3. Existence of inverse : ∀ u ∈ V , there is a vector in V , denoted by -u and called the negative of u , such that u + (-u) = (-u) + u = 0 .
4. Commutative property : u + v = v + u ∀ u , v ∈ V .
5. k(u + v) = ku + kv ∀ u , v ∈ V and k ∈ K .
6. (a + b)u = au + bu ∀ u ∈ V and a , b ∈ K .
7. (ab)u = a(bu) ∀ u ∈ V and a , b ∈ K .
8. 1u = u ∀ u ∈ V where 1 ∈ K is the unity .

Alternative definition :

(V , +) be an algebraic structure and (F , + , •) be a field , then V is called a vector space over the field F if the following conditions hold :

1. (V , +) is an abelian group .
2. ku ∈ V ∀ u ∈ V and k ∈ F
3. k(u + v) = ku + kv ∀ u , v ∈ V and k ∈ F .
4. (a + b)u = au + bu ∀ u ∈ V and a , b ∈ F .
5. (ab)u = a(bu) ∀ u ∈ V and a , b ∈ F .
6. 1u = u ∀ u ∈ V where 1 ∈ F is the unity .

♦ Elements of V are called vectors .

♦ Elements of F are called scalars .

♦ If V is a vector space over the field F then it is denoted by V(F) .