Question

Let V = {p(t) :p is a polynomial} over the field R. Define the inner product, (p,q)= ∫²₀ p(t)q(t)dt. For the following linear maps on V determine whether they are self-adjoint, (a) Lp(t) = (t²+2)p(t). (b) Lp(t) = p(-t).

Answer 1

To determine whether a linear map is self-adjoint, we need to check if (Lp,q) = (p,Lq) for all polynomials p and q in V. For Lp(t) = (t²+2)p(t), both sides of the equation are equal, so it is self-adjoint. For Lp(t) = p(-t), the integrals are not guaranteed to be equal, so it is not self-adjoint.

Step-by-step explanation:

To determine whether a linear map is self-adjoint, we need to check if (Lp,q) = (p,Lq) for all polynomials p and q in V.

(a) For Lp(t) = (t²+2)p(t), let's calculate both sides of the equation:

(Lp,q) = ∫²₀ (t²+2)p(t)q(t)dt = ∫²₀ t²p(t)q(t)dt + 2∫²₀ p(t)q(t)dt

(p,Lq) = ∫²₀ p(t)(t²+2)q(t)dt = ∫²₀ p(t)t²q(t)dt + 2∫²₀ p(t)q(t)dt

Since both sides of the equation are equal, Lp(t) = (t²+2)p(t) is self-adjoint.

(b) For Lp(t) = p(-t), let's calculate both sides of the equation:

(Lp,q) = ∫²₀ p(-t)q(t)dt

(p,Lq) = ∫²₀ p(t)q(-t)dt

Since the integral of p(-t)q(t) is not guaranteed to be equal to the integral of p(t)q(-t), Lp(t) = p(-t) is not self-adjoint.