Question

Prove that if V and W are both subspaces of the vector space U, then their intersection is also a subspace of U. Please write a formal proof and write legibly! Show all steps. Linear Algebra. Thanks.

Answer 1

Answer: The proof is done below.

Step-by-step explanation: We are given to prove the following statement :

If V and W are both sub spaces of the vector space U, then their intersection is also a subspace of U.

According to the definition of a subspace, we can say that {0} belongs to both V and W.

So, {0} will also belong to the intersection of V and W.

That is, {0} ∈ V ∩ W.

Now, let a, b are scalars and v, w∈ V ∩ W.

So, we get

v, w ∈ V and v, w ∈ W.

Since V and W are sub spaces of V and W, so we get

av + bw ∈ V and av + bw ∈ W.

Therefore, av + bw ∈ V ∩ W.

Thus, V ∩ W is also a subspace of U.

Hence proved.