We let $\mathcal{B} = \{\alpha_1, ... \alpha_n\}$ be a basis for $V$, a vector space with inner product $\langle \cdot, \cdot \rangle$. Then we define $f_i(v) := \langle v, \alpha_i \rangle$.
Show that $f_1, ..., f_n$ is a basis for $V^{\star}$ the dual space of $V$.
This was a question on a midterm I just finished, I honestly had no idea how to do it here's my relevant work:
I know that if $f_1, ..., f_n$ are independent than they are a basis since $\dim(V^{\star}) = \dim(V) = n$ because they are finite dimensional. So I have to show that they are independent, this is where I am totally lost.
I can show that if $\mathcal{B}$ is orthogonal then they are linearly independent, since if $c_1f_1 + ... + c_nf_n = 0$ implies we must have $(c_1f_1 + ... + c_nf_n)(\alpha_i) = 0$ for each $\alpha_i$ and since the $\alpha_i$ are orthogonal then for each $i$ this becomes: $c_i \langle \alpha_i, \alpha_i \rangle = 0$ but $\langle \alpha_i, \alpha_i \rangle \neq 0$ since $\alpha_i \neq 0$ and hence $c_i = 0$ for all $i$.
Now I wonder is there a way to generalize an approach like this to prove the actual question posed here, or am I going about this completely wrong? If the second is the case, how should I have gone about solving this problem?
Thanks,