1. Let C be a set of vectors in some vector space V, and let M be some subspace of V. Show that if C C M then (C) C M. (This shows that (C) is the smallest subspace containing C.) 2. Recall the vector space V= : a, b E R, a, b > 0 ) over the field R, with vector addition defined , and scalar multiplication as t as " + [a] - [oa]. and scalar = bt a Find a basis for V. b) Show that V ~ R" for some n. Explicitly give an isomorphism. Prove your answer! 3. Suppose that V = MON and that {x1, . ..,x} is a basis for M and {y1, . .. ,ys} is a basis for N. Show that {X1, . .., Xr, y1, . .. ,ys} is a basis for V. 4. Suppose that {X1, X2, . ..,Xn} is a basis for V. For some 1 _ r _ n let M = (x1, . .. ,X,) and N = (Xr+1, . .. ,Xn). Show that V = MON. 5. Suppose that {X1, X2, X3, x4} is a basis for a vector space V. Suppose that U is a subspace of V, and that x1, X2 E U but x3 @ U and x4 $ U. Is it true that {x1, x2} is a basis for U? Proof or counterexample