Scalar product in a space of polynomials

Question

x−1,2x−1 create a perpendicular and normalised base in a space of polynomials of a degree fewer or equal to one with some scalar product. Find the pattern of this scalar product.

Answer 1

Hint:

use the fact hat the vector space of the polynomials $P^{(1)}(\mathbb{R})$ is isomorphic to $\mathbb{R}^2$ as a vector space over $\mathbb{R}$ with the correspondence: $$ a+bx\qquad \leftrightarrow \qquad (a,b)^T $$