Let H and K be subspaces of a vector space V. The intersection of H and K,

Question:

Let H and K be subspaces of a vector space V. The intersection of H and K, written as H ( K, is the set of v in V that belong to both H and K. Show that H ( K is a subspace of V. (See the figure.) Give an example in R2 to show that the union of two subspaces is not, in general, a subspace.