Finding a dual basis for a space of polynomials with a given degree bound I am looking over Linear Algebra,and am stuck on this problem.   V is the space of real polynomials of degree 2 or less, and I want to find a dual basis to the basis $\{1,x,x^2\}$. I know the general technique, that each element of the dual basis $e_i'$evaluates to $0$ on the basis vectors $e_j$ for which $i \neq j$ and $1$ when $i=j$ but every time I try and do this on these basis vectors, I get stuck.   Any help appreciated.   Also, a more general question: what is the method for determining in general whether or not something is a dual basis? I know you have to show it to be linearly independent and spanning, and showing linear independence is generally quite easy, but how do I show spanning? If I have a dual basis I'm presuming it doesn't necessarily have to satisfy the above criterion (evaluating to 0 and 1 on different basis vectors), and this is just a way of finding one dual basis, so how would I test whether or not a given dual basis is spanning?   Thanks.
Finding the dual basis for the space of polynomials of degree $\leq$ 2 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.
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