Question

union of two subspaces of a vector space is a subspace of a vector space. (true/false)

Answer 1

The reason why this can happen is that all vector spaces, and hence subspaces too, must be closed under addition (and scalar multiplication). The union of two subspaces takes all the elements already in those spaces, and nothing more. In the union of subspaces W1 and W2, there are new combinations of vectors we can add together that we couldn't before, like v1+w2 where v1∈W1 and w2∈W2.
For example, take W1 to be the x-axis and W2the y-axis, both subspaces of R2.
Their union includes both (3,0) and (0,5), whose sum, (3,5), is not in the union. Hence, the union is not a vector space.

for this reason it is false

Answer 2

Answer :

False

Explanation :

Union of two subspaces of a vector space is not always a subspace .

Union of two subspaces of a vector space is a subspace iff one of them is contained in other .

(Please refer to the attachment for proof)

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 .