How to find the corresponding matrix of a dot product over a polynomial ring to a specific basis

Question

Let $V= \mathbb R[x]_{\leq 2}$ be the vector-space of real polynomials with degree $\leq 2$. We define a dot product on the $V$ as follows: $$\left<f,g \right> = \int_{0}^1f(x)g(x)dx.$$

Question: What is the corresponding matrix $A := ( \left< b_i,b_j \right> )_{i,j}$ with respect to the basis $B=\{1,x,x²\}$ of $V$?

My approach until now looks like this:
$\int_0^1 (ax²+bx+c)(dx²+ex+f) dx$

Answer 1

You seek a $3\times 3$ matrix $$ A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} $$ The formula for $a_{ij}$ is $$ a_{ij}=\int_0^1 e_i(x)\cdot e_j(x)\,dx $$ where \begin{align*} e_1(x) &= 1 & e_2(x) &= x & e_3(x) &= x^2 \end{align*} For example, $$ a_{11} = \int_0^1 e_1(x)\cdot e_1(x)\,dx=\int_0^11\cdot1\,dx=\int_0^1\,dx=1 $$ Can you compute the other eight entries?

Answer 2

All we have to do is evaluate the $9$ integrals ($5$ really). You don't have to try and find the general form of the inner product to compute $A$. The basis elements are all from the set $\{1,x,x^2\}$, so the integrals you need to evaluate are

$$\left[\begin{matrix}\langle 1,1\rangle=\int\limits_0^11\,dx&\langle 1,x\rangle=\int\limits_0^1x\,dx&\langle 1,x^2\rangle=\int\limits_0^1x^2\,dx \\\langle x,1\rangle=\int\limits_0^1x\,dx&\langle x,x\rangle=\int\limits_0^1x^2\,dx&\langle x,x^2\rangle=\int\limits_0^1x^3\,dx \\\langle x^2,1\rangle=\int\limits_0^1x^2\,dx&\langle x^2,x\rangle=\int\limits_0^1x^3\,dx&\langle x^2,x^2\rangle=\int\limits_0^1x^4\,dx\end{matrix}\right]$$