Let $p$ and $q$ be two quadratic polynomials, given by $$ p(x)=c_1+c_2x+c_3x^2, \qquad q(x)=d_1+d_2x+d_3x^2 $$ Express the integral $J=\int_0^1 p(x)q(x)\,dx$ in the form $J=c^TGd$, where $G$ is a $3\times 3$ matrix. Give the entries of $G$ (as numbers).
In this particular question, what will the equation for $J$ be, if I had help on that I could figure out the $3\times 3$ matrix $G$.
p.s. my lack of understanding of calculus limits me. Thank you.
If we consider the vector space $V$ of all polynomials of degree at most $2$, then the bilinear form $$ \langle p(x),q(x)\rangle=\int_0^1 p(x)q(x)\,dx $$ is an inner product on $V$. The matrix $G$ is the Gramian matrix of this inner product with respect to the basis $\{1,x,x^2\}$. If $f_1(x)=1$, $f_2(x)=x$, $f_3(x)=x^2$, then the Gram matrix has as $(i,j)$ entry the inner product $$ \langle f_i(x),f_j(x)\rangle $$ Now it's computation time.