Problem 5. (4+3+3) (i) Let A be a square matrix of order n and u a column vector in R" such that A³ = Onxn and A²u0. Show that u, Au,

A²u are linearly independent. (ii) Find three independent vectors on the solution space of the equation x + 2y - 3z - t = 0 in R4. Can you find four? (iii) For any two subspaces V, W of R³, we define the set V + W = {x € R³ | there exist v € V and w€ W such that x = v+w}. It is then known that V+W is also a subspace. Assume now that V = {(t, 2t, 3t) | t = R}, W = {(t, -2t,0) | t ≤ R}. = span(S). Find a finite set S such that V+W =