Prob1. LetTL(V,W).Prove (a) T is injective if and only if T is surjective;

(b) T is injective if and only if T is surjective.

Prob 2. Suppose S, T L(V ) are self-adjoint. Prove that ST is self-adjoint if and only if ST = T S.

Prob3. LetPL(V) be such thatP2=P. Prove that there is a subspaceUofVsuch thatPU=Pifand only if P is self-adjoint.

Prob4. LetnIN be fixed. Consider the real spaceV :==span(1,cosx,sinx,cos2x,sin2x,...,cosnx,sinnx)with inner product

f, g :=f(x)g(x)dx (from -pi to pi)

Show that the differentiation operator D L(V ) is anti-Hermitian, i.e., satisfies D = D.

Prob 5. Let T be a normal operator on V . Evaluate T (v w) given that Tv = 2v, Tw = 3w, v = w = 1.

Prob 6. Suppose T is normal. Prove that, for any IF, Null(T I)^k =Null(T I).