A linear transformation T Rn rightarrow Rn is called an idem

A linear transformation T: R^n rightarrow R^n is called an idempotent if T^2 = T. Give an example of an idempotent T such that T notequalto Id and T notequalto 0. Let T be an idempotent. Prove that ker(T) + im(T) = R^n. Let T be an idempotent. Prove that Tv vector = v vector for any v vector elementof im(T). Conclude that there exist subspaces V_0 and V_1 of R^n such that V_0 + V_1 = R^n and for every V_0 elementof notequalto V_0 and v_1 vector elementof V_1, Tv_0 vector = 0 vector and T v_1 vector = v_1 vector.

Solution

(a) consider the function that assigns to every subset S of a topological space X, the closure of U is an idempotent on the power set P(X) of X.

(b)