Let $T:\mathbb{P_2\to R}$ be transformation. Find basis for kernel of $T$ that is orthogonal w. r. t. inner product. $⟨f(x),g(x)⟩=\int_0^1f(x)g(x)dx$

Question

Problem

• Let $T : \mathbb{P_2\to R}$ be the function defined by $T(p(x))= p(1)$. Verify that $T$ is in fact a linear transformation. (There are two things you need to verify for this.)
• Let $T$ be the transformation defined above. Find a basis for the kernel of $T$ that is orthogonal with respect to the inner product.$$ \langle f(x),g(x)\rangle =\int_0^1f(x)g(x)\mathrm{dx}$$ (That is, find a basis for the kernel of $T$ such that all pairs of distinct basis vectors are orthogonal with respect to this inner product.)

What is the question asking in the second part? How to find a basis for the kernel of $T$ that is orthogonal with respect to an inner product?