Question

linear sum of two subspaces of vector space is also a subspace​

Answer 1

Step-by-step explanation:

The Intersection of Two Subspaces is also a Subspace Let U and V be subspaces of the n-dimensional vector space Rn. Prove that the intersection U∩V is also a subspace of Rn. ... The Subspace of Linear Combinations whose Sums of Coefficients are zero Let V be a vector space over a scalar field K.

Answer 2

To prove :

The sum linear sum of two subspaces is a subspace .

Proof :

Please refer to the attachment .

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 .

Linear sum :

If W₁ and W₂ are two subspaces of the vector space V(F) , then the sum W₁ + W₂ = {w₁ + w₂ : w₁ ∈ W₁ , w₂ ∈ W₂} is called the linear sum of W₁ and W₂ .