A subspace is a subset of a vector space that satisfies all the properties of a vector space. This means that it is closed under vector addition and scalar multiplication, and contains the zero vector.
The intersection of two subspaces is the set of all elements that are common to both subspaces. In other words, it is the subset that is contained in both subspaces.
To prove that the intersection of two subspaces H and K is a subspace of vector space V, we need to show that it satisfies the three properties of a subspace: closure under vector addition, closure under scalar multiplication, and contains the zero vector.
Proving that the intersection of two subspaces is a subspace is important because it allows us to use the properties of a vector space on this subset. This can be helpful in solving problems and proving other theorems related to vector spaces.
Yes, it is possible for the intersection of two subspaces to be empty. This would occur if the two subspaces do not have any elements in common, meaning they are linearly independent and do not share any vectors.