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:  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 getv, w ∈ V  and  v, w ∈ W.Since V and W are sub spaces of V and W, so we getav + bw ∈ V  and  av + bw ∈ W.Therefore, av + bw ∈ V ∩ W.Thus, V ∩ W is also a subspace of U.Hence proved.