I have been struggling with the following question:
Suppose that $X$ is n-dimensional, let ${e_1,...,e_n}$ be an orthonormal basis of $X$ and suppose that $\phi : X \rightarrow \mathbb{C}$ is a linear map. The Reisz Representation Theorem guarantees that there is some $y \in X$ such that $\phi(x) = \langle y,x \rangle$ for all $x \in \mathbb{X}$. The fact that ${e_1,...,e_n}$ is a basis means that $y = \sum_{i=1}^{n} \alpha_i e_i$ for some choice of $\alpha_i \in \mathbb{c}.$ written in terms of $\phi$ and the basis vectors $e_i$.Find an expression for the coefficients $\alpha_{i}$ written in terms of $\phi$ and the basis vectors.
All I can think to do is the following:
$\phi(x) = \langle y,x \rangle = \langle \sum_{i=1}^{n} \alpha_i e_i , x \rangle $
I don't know where to go from here, so any help would be greatly appreciated.
You are on the right track, but you can also expand $x$ in terms of the orthonormal basis as well. This gives, $\phi(x) = \langle y,x\rangle = \langle \sum\limits_{i=1}^n \alpha_ie_i,\sum\limits_{j=1}^n\beta_je_j\rangle = \sum\limits_{i=1}^n\alpha_i\langle e_i,\sum\limits_{j=1}^n\beta_je_j\rangle = \sum\limits_{i=1}^n\alpha_i\overline{\langle\sum\limits_{j=1}^n\beta_je_j,e_i\rangle}=\sum\limits_{i=1}^n\alpha_i \left(\sum\limits_{j=1}^n\overline\beta_j\langle e_i,e_j\rangle\right)$
Now we use the orthonormality of the $e_n$ to get, $\phi(x) = \sum\limits_{i=1}^n\alpha_i\overline\beta_i$