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.
