Exercise 17

Consider the isomorphism $\text{Rep}_{B}\colon\mathcal{P}_{3}\to\mathbb{R}^{4}$.

• Vectors in a real space are orthogonal if and only if their dot product is zero. Give a definition of orthogonality for polynomials.

• The derivative of a member of $\mathcal{P}_{3}$ is in $\mathcal{P}_{3}$. Give a definition of the derivative of a vector in $\mathbb{R}^{4}$.