Question

Let α, β, γ ∈ ℝ designate pairwise different real numbers and understand the ℝ-vectorspace P₃(ℝ) of real polynomials of degree 2 or less as an inner product space via. = p(α)q(α) + p(β)q(β) + p(γ)q(γ). Now let λ ∈ C / ℝ designate a complex number which is NOT a real number.

Question: Show that for every p, q ∈ P₃(ℝ) it holds that is a real number. (Hint: show that the number doesn't change through complex conjugation. (NOTE: The notation designate the complex conjugation of .

Please provide explanations for every step as I learn alot from those. Many thanks in advance and have a blessed day.

Answer 1

As, lambda elementof C - R lambda notequalto 0. As, p(lambda), p(lambda)^bar, q(lambda), q(lambda)^bar elementof C Let f(lambda) = p(lambda) q(lambda)^bar + p(lambda)^bar q(lambda) then, f(lambda)^bar = p(lambda) q(lambda)^bar + p(lambda)^bar q(lambda) ^bar = p(lambda) q(lambda)^bar ^bar + p(lambda)^bar q(lambda) ^bar = p(lambda)^bar q(lambda) + p(lambda q(lambda)^bar = f(lambda) So, f(lambda) is areal number. Also, p(1) q(1) is a real number So, p(lambda) q(lambda)^bar + p(1) q(1) + p(lambda)^bar q(lambda) is a real number.