Question

The intersection of any two subspaces of a vector space
a subspace

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Answer 1

Answer:

Step-by-step explanation:

Let M, N be two sub-spaces of a vector-space V(F) .

Let W = MnN .

To prove W a sub-space of V

one must show that (ax +by)€ W,    whenever x, y € W and a, b € F .

clealy , if x, y € W = MnN ==> x, y €M &

x, y € N ==> (ax + by)€M & (ax + by)€ N

for any two scalars a, b belonging to

F ==>

         (ax + by) € MnN = W . This proves the requirement.

Answer 2

To prove :

Intersection of two subspaces of a vector space is a subspace .

Proof :

Let W₁ and W₂ be two subspaces of the vector space V(F) .

Let x , y ∈ W₁∩W₂ , then

→ x , y ∈ W₁ and x , y ∈ W₂

Now ,

Since W₁ is a subspace of V(F) , then

x , y ∈ W₁ → ax + by ∈ W₁ ∀ a , b ∈ F .

Also. ,

Since W₂ is a subspace of V(F) , then

x , y ∈ W₂ → ax + by ∈ W₂ ∀ a , b ∈ F .

Now ,

Since ax + by ∈ W₁ and ax + by ∈ W₂ , thus

→ ax + by ∈ W₁∩W₂ ∀ x , y ∈ W₁nW₂ and a , b ∈ F

→ W₁∩W₂ is a subspace of V(F) .

Hence proved .

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) .

Subspace :

A non empty subset W of the vector space V(F) is said to be a subspace of V if it itself forms a vector space over the same field F .

♦ A non empty subset W of V is said to be a subspace of V(F) iff ax + by ∈ W for every a , b ∈ F and x , y ∈ W .