Let U and V be subspaces of a vector space W. The sum of U and V, denoted U + V, is defined to be the set of all vectors of the form u + v, where u U and v V. Prove that U + V and U V are subspaces of W. If U + V = W and U V = 0, then W is said to be the direct sum. In this case, we write W = U V. Show that every element w W can be written uniquely as w = u + v, where u U and v V. Let U be a subspace of dimension k of a vector space W of dimension n. Prove that there exists a subspace V of dimension n - k such that W = U V. Is the subspace V unique? If U and V are arbitrary subspaces of a vector space W, show that
