Let W. Z V be complementary subspaces in a vector space V, as in Exercise 2.2.24.

Question:

Let W. Z ⊂ V be complementary subspaces in a vector space V, as in Exercise 2.2.24.
(a) Prove that if (w1,...,wj} form a basis for W and {z1.......zk} a basis for Z, then {w1,..., wj, z1,.... zk} form a basis for V.
(b) Prove that dim W + dim Z = dim V.