Let V be a vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T). Prove that the map (ST)² is the zero map.

Let I be the line in R³ that consists of all scalar multiples of the vector |2 Find the orthogonal projection of the 2 9 vector x = 8 onto L. projL =

2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.

Let p(x) = a, +a,x+ a,x* +a,x' and g(x) = 6, + 6x+ 6,x + b,x' be vectors in P, with the standard inner product space on P,defined by = a,b, + a,b, + a,b, + a,b;. a) Find the distance between fand g f - | angle of f(x) and h(x) b) Use the Gram-Schmidt process to form an orthogonal set for the functions f(x) = 1-x-x* + 2x', g(x) = -2+ 2x - x and h(x) = x - x* + 2x