Reference no: EM13863414

Prob1. LetT∈L(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. Let P∈L(V) be such that P2=P. Prove that there is a subspace U of V such that PU=P if and only if P is self-adjoint.

Prob4. Let n∈IN be fixed. Consider the real space V :==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 λ ∈ IF,
Null(T -λI)^k =Null(T -λI).