A basis $\{p_1, p_2, p_3\}$ of Polynomial space of degree $2$, such that $[t^2]$ of the basis is $(1, 1, 0)$

Question

The only basis that I know of in $P_2$ is ${1, t, t^2}$, however that would not yeild $(1, 1, 0)$ for $t^2$, instead it would be $(0, 0, 1)$. How can I come up with this basis, so that for $t^2$ I would get $(1, 1, 0)$?

Answer 1

We want (1): $\ t^2=p_1+p_2$.

We have many possibilities for choosing such a basis $p_1,p_2,p_3$.

Probably the easiest is if we take $p_1:=t^2-t$ and $p_2:=t$.
This certainly satisfies (1), which already ensures that $t^2$ is in the span of $p_1,p_2(,p_3)$.
Taking $p_3:=1$, we see that all standard basis elements, hence all polynomials of degree $\le2$, are generated by $p_1,p_2,p_3$, so it is a basis of this $3$ dimensional space.