Every $n$-tuple of scalars $(\alpha_1, \dots, \alpha_n)$ determines a linear functional on a finite dimensional normed space $X$.
What is the proof of this? Why does there exist a linear functional $f : X \rightarrow \Bbb R$ such that $f(e_i) = \alpha_i$ for each $n$-tuple? What are the functionals? How can linearity be preserved with every choice of $\alpha$'s?
I can see if $x = \sum_i c_i e_i$ then $f(x) = \sum_i c_i f(e_i) = \sum_i c_i \alpha_i$ where $\alpha_i = f(e_i)$, but I don't understand how this can be reversible to start with an $n$-tuple and find a functional.
Determining the values of a linear map $T:V\rightarrow W$ on the basis elements of $V$ determines $T$, by the definition of a basis for a vector space. Your case is when $W=\mathbb{R}$.
This is well defined given that you are working with a fixed basis for all functionals, and not changing basis. For a different basis, this will give you different functionals.