The following is from the book "Tensor Methods in Statistics", which could be downloaded there

http://www.stat.uchicago.edu/~pmcc/tensorbook/

I have a question regarding section 0.3.1, titled "Duality and dual spaces", there it is said

Let $V$ be a vector space with basis $\{ e_1, \ldots, e_n \}$. In the usual expression for $v \in V$ as a linear combination of the basis vectors, we write $v = v^i e_i$. The notation exhibits a certain formal symmetry between the components $v^i$ and the basis vectors $e_i$, but the interpretation, as usually given, is quite asymmetric: $v^i$ is a 'scalar', whereas $e_i$ is a 'vector' in $V$. However $\{v^i, \ldots, v^n\}$, as linear functionals, form a basis in $V^*$.

Here comes my first question, how could the scalar $v^i$ be regarded as a linear functional, and furthermore, how could $\{ v^i, \ldots, v^n\}$ be interpreted as a basis for $V^*$ when its just a set of scalars?

He continues

Thus, the expression $v^i e_i$ could equally well be interpreted as a point in $V^*$ with components $(e_1, \ldots, e_n)$ relative to the dual basis.

Now he interprets the basis vectors $e_i$ as components in the dual basis, but I thought linear functionals, i.e. element of $V^*$ could be represented equally by $n$-tuples of numbers.