Problem set 4 Math 110 Lec 2 Due: Wed. 7/20 1,2 can be solved after 7/11. 3,4 can be solved after 7/12. 5,6 can be solved after 7/13. 7,8 can be solved after 7/14. 1. (Axler 3.F.14) Define T : P ( R ) → P ( R ) , ( Tp )( x ) := x 2 p ( x ) + p 00 ( x ) . (i) Define ϕ ∈ P ( R ) 0 by ϕ ( p ) := p 0 (4) . Describe T 0 ( ϕ ). (ii) Define ϕ ∈ P ( R ) 0 by ϕ ( p ) := R 1 0 p ( x ) d x. Evaluate ( T 0 ϕ )( x 3 ) . 2. Let V be a vector space. (i) Prove that T : V 0 × V 0 -→ ( V × V ) 0 , T ( ϕ, ψ )( v, w ) := ϕ ( v ) + ψ ( w ) is an isomorphism. (ii) Consider the diagonal map Δ : V → V × V, v 7→ ( v, v ) . After identifying ( V × V ) 0 with V 0 × V 0 as in (i), the dual map becomes a map Δ 0 : V 0 × V 0 -→ V 0 . Describe this map. 3. (Axler 5.A.26, 5.A.27) Suppose V is finite-dimensional and T ∈ L ( V ). (i) Assume that every nonzero vector in V is an eigenvector of T . Prove that T is a scalar multiple of the identity operator. (ii) Assume that every hyperplane (subspace U ⊆ V of dimension dim( U ) = dim( V ) - 1) is invariant under T . Prove that T is a scalar multiple of the identity operator. Continued on the back
4. (Axler 5.A.35, 5.A.36) Let T ∈ L ( V ) be an operator and U ⊆ V be an invariant subspace. Let T denote the quotient operator T ∈ L ( V/U ) , T ( v + U ) := T ( v ) + U. (i) Assume that V is finite-dimensional. Prove that every eigenvalue of T is also an eigen- value of T . (ii) Give a counterexample to show that when V is infinite-dimensional, T can have an eigen- value that is not an eigenvalue of T . 5. (Axler 5.B.20) Suppose V is a finite-dimensional complex vector space and T ∈ L ( V ).