(a) Prove that if v1,..., vm forms a basis for V Rn, then m < n....

Question:

(a) Prove that if v1,..., vm forms a basis for V ⊆ Rn, then m < n.
(b) Under the hypotheses of part (b). prove that there exist vectors vm+1. . .,vn e Rn \ V such that the complete collection v1,..., vn forms a basis for Rn.
(c) Illustrate by constructing bases of R3 that include
(i) the basis (1, 1,1/2)T of the line x = y = 2z:
(ii) the basis (1,0, -1)T ∙ (0,1,-2)T of the plane x + 2y + z = 0