Let $V$ be a Vector Space of polynomials of degree less than or equal to n. Define $\alpha_{k}: V \rightarrow \mathbb{R}$ by $\alpha_{k}(p)=\int_{-1}^1$ ${t^{k}}p(t)dt, p\in V$
Show that $\{{\alpha_{0},\alpha_{1}...\alpha_{n}}\}$ is a basis for the dual space of V.
I have a hint that $dimD(V)=n+1$ and so I only need to prove linear independence.
Thought process: Let $B$ be the basis for $V$ s.t $B=\{1,x,x^{2},...x^{n}\}$ This is a linearly independent set, $\sum_{i=0}^{n}$b_{i}x_{i}$ $\forall b \in F$, and has dimension of $n+1$. I need to show that $\sum_{i=0}^{n}$a_{i}\alpha_{0}=0$ $\forall a \in F$ Applying $\alpha$ to each vector gives us 0 each time by the way the integral is constructed (assuming I evaluated p=1 correctly and generalizing), hence $\sum_{i=0}^{n}$a_{i}\alpha_{0}=\sum_{i=0}^{n}$b_{i}x_{i}$$=0$
Is this decent? Also somewhat unrelated, but should this integrand remind me of the Laplace Transformation?