Let $\{v_i\}_{i=1}^n$ be an orthonormal basis for the $n$-dimensional vector space $\mathcal V$. Let $x\in \mathcal V$ have the representation $x=\sum b_i v_i$. Show that the Fourier coefficients $b_i$ can be computed as $b_i=\langle x, v_i\rangle$.
If I take the inner product of both sides I get $$\begin{split}\langle x, v_i\rangle&=\langle \sum_{j=1}^n b_j v_j, v_i\rangle\\ &=\sum_{j=1}^n b_j\langle v_j, v_i\rangle\\ &=b_i\|v_i\|^2\\ &=b_i\end{split}$$
Is the point of this that $x$ might not be written in the form $\sum b_iv_i$ explicitly (i.e. you can't actually see the $b_i$'s) and so you can find the $b_i$'s by taking the inner product w.r.t. each orthonormal basis vector? What I'm trying to say is that the way it's written, I can see the $b_i$'s. Am I supposed to assume that I don't know them or is it actually about you can see the $b_i$'s and you can also get them by doing the inner product. (Or both?)
Such linear combination always exists, no matter whether you can see it immediately or not. Since it exists, it makes sense for us to write this expression.
And yes, you are correct that the coordinate $b_i$'s can be found by taking the inner product between $x$ and the basis element $v_i$'s though we may not know it before.