Let U and W be subspaces of a vector space V. We say that V is the direct sum of U and W, written V=U⊕W, if every vector v∈V can be written uniquely as a sum v=u+w with u∈U and w∈W. In this problem we will assume that V is finite-dimensional. (a) Prove that if V=U⊕W then U∩W={0} and dim(V)=dim(U)+dim(W). (b) Suppose that V is an inner product space and let W be a subspace of V. Show that V=W⊕W⊥ (c) Prove/disprove: If V is a finite-dimensional inner product space, if U and W are subspaces of V, and if V=U⊕W, then W=U⊥.
