Let V be a vector space. If f: V → [R is a linear transformation and z is a vector in V, define Tj z: V → V by Tf.z(v) = f(v)z for all v in V. Assume that f ≠ 0 and z ≠ 0.
(a) Show that Tj z is a linear operator of rank 1.
(b) If ≠ 0, show that 7f . z is an idempotent if and only if f(z) = 1. (Recall that T: V → V is called an idempotent if T2 = T.)
(c) Show' that every idempotent T: V→ V of rank 1 has the form T = Tf z for some f : V→R and some z in V with f(z) = 1. [Hint: Write im T = IRz and show that T(z) = z. Then use Exercise 23.]