vector space

[Cesarruletita]

Let [V] a vector space over the field [F] and let [g] the polynomial on [F] given by [g (x) = a_0 + ... + a_nx a_1x + ^ n]. For each operator [T] on [V] defines a transformation

 

[G (t): V -----> V] as [g (t) = a0 I+ a_1T +......+ a_nT ^ n]

 

a) Prove that [g (T)] is an operator on [V]

 

b) Let [a = a_0 + a_1 + ... + a_n] and let [E] a projection of [V]. Prove that if [n] is an even integer if [a = a_0], then [g (E)] is a scalar multiple of the identity operator.

Edited November 18, 2012 by Cesarruletita

[ydoaPs]

Your relation G(t) is mapping V onto V?