Question

whats a quotient space?​

Answer 1

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).

Answer 2

Some important information :

Vector space :

(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 and the lements of F are called scalars .

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

Answer :

Quotient space :

Let W be a subspace of V(F) , then V/W = {x + w : x ∈ V , w ∈ W} is called the quotient space .

How addition is defined in quotient space :

Let (x + w) , (y + w) ∈ V/W , where x , y ∈ V and w ∈ W , then (x + w) + (y + w) = (x + y) + w ∈ V/W .

How scalar multiplication is defined in quotient space :

Let x + w ∈ V/W where x ∈ V and w ∈ W .

Let a ∈ F , then a(x + w) = ax + w ∈ V/W .

♦ Clearly , the addition and the scalar multiplication are closed .