Let U be an m-dimensional subspace of Rn and let V be a k-dimensional subspace of U,

Question:

Let U be an m-dimensional subspace of Rn and let V be a k-dimensional subspace of U, where 0 < k < m.
(a) Show that any orthonormal basis (v1, v2,...,vk} for V can be expanded to form an orthonormal basis {v1, v2,..., vk, vk+1,..., vm} for U.
(b) Show that if W = Span(v1 +.....+ vk+2 ..., vm), then U = V ⊕ W.