Q1: Let $V = \mathcal{P}2( [-1,1], \mathbb{R} )$ be the polynomial space with inner product $ \langle f, g \rangle = \int{-1}^{1} f(x) g(x) , dx $. Is the monomial basis $(m_0, m_1, m_2)$ an ONB?

A1: No, it’s not orthogonal.

A2: Yes, it is.

A3: One needs more information.

Q2: Let $V = \mathcal{P}2( [-1,1], \mathbb{R} )$ be the polynomial space with inner product $ \langle f, g \rangle = \int{-1}^{1} f(x) g(x) , dx $. What do we get when we apply the Gram-Schmidt procedure to the monomial basis $(m_0, m_1, m_2)$?

A1: We get an ONB of polynomials, called the Legendre polynomials.

A2: We get a basis $(b_1, b_2, b_3)$ with $b_1 = m_0$.

A3: We get an OB $(b_1, b_2, b_3)$ with $b_1 = \frac{1}{2} m_0$.