Let W ⊂ V be a subspace. A subspace Z ⊂ V is called a complementary subspace to W if
(i) W ∩ Z = {0}, and
(ii) W + Z = V, i .e., every v ∈ V can be written as v = w + z for w ∈ W and z ∈ Z.
(a) Show that the .v and y axes are complementary subspaces of R2.
(b) Show that the lines x = y and x = 3y are complementary subspaces of R2.
(c) Show that the line (a, 2a, 3a)T and the plane x + 2y + 3z = 0 are complementary subspaces of R3.
(d) Prove that if v = w + z then the summands w ∈ W and z ∈ Z are uniquely determined.