Let $V$ be the space of polynomials of degree at most $2$, with basis $\{1,x,x^2 \}$.Find the dual basis to $\{1,x,x^2\}$.
I know that the dual basis of $\{f_1,f_2,f_3\}$ is the basis $\{f_1^*,f_2^*,f_3^*\}$ where $f_i^*(a_1f_1,\ldots,a_3f_3)$ but I'm not sure how to actually compute the dual basis.
Best Answer
A dual basis functional acting on a polynomial would extract the coefficient at the corresponding monomial. The natural candidate for the dual basis therefore is the linear differential operators $D^k=\tfrac1{k!}\tfrac{d^k} {dx^k}\big|_{x=0}$, $k=0,1,2,..$ -- the familiar formula for the coefficients of the Taylor (Maclaurin) series.