TITLE: Trouble with understanding dual space $V^{*}$.
QUESTION [5 upvotes]: I've just began the study of linear functionals and the dual base. And this book I'm reading says the dual space $V^{*}$ may be identified with the space of row vectors. This notion seems very important, but I'm having trouble understanding it. Here is the text:

Let $\sigma$ be an element of the dual space $V^{*}$, i.e. a linear
  map $\sigma: V \rightarrow K$. Choose a basis for $V$, say the usual
  the basis, then $\sigma$ is represented by a matrix $[\sigma]$.
  However, such a matrix $[\sigma]$ is a row vector. Also, the map
  $\sigma \rightarrow [\sigma]$ is a vector space isomorphism.
On the other hand, any row vector $\phi = (a_1, \ldots, a_n)$ defines
  a linear functional $\phi: V \rightarrow K$ by \begin{align*}
\phi(x_1, \ldots, x_n) = (a_1, \ldots, a_n) \begin{pmatrix} x_1 \\
\vdots \\ x_n \end{pmatrix} \end{align*} or simply $\phi(x_1, \ldots,
 x_n) = a_1 x_1 + a_2 x_2 + \ldots + a_n x_n$.

The author speaks of the matrixrepresentation $[\sigma]$, but he doesn't really explain it. Why is this matrix a row vector? Also, the second part of the text: is this merely a definition? Why does he claim $\phi(x_1, \ldots, x_n) = a_1 x_1 + \ldots + a_n x_n$? The output of a linear functional is suppose to be a scalar, and not a vector? And this is clearly a linear combination of vectors...
Maybe some of the advanced mathematicians here could give me some examples, because I can't get my head around this at the moment.

REPLY [2 votes]: The elements of $V^*$  operate on $V$ in the obvious way: $(\varphi ,v)\mapsto \varphi (v)$.
We ask how can we represent this using the bases for $V$ and $V^*$.
Given the basis $\mathcal B=\left \{ v_{1},\cdots ,v_{n} \right \}$ for $V$, there is the natural basis $\mathcal B^*=\left \{ v_{1}^*,\cdots ,v_{n}^* \right \}$ for $V^*$ defined by 
$v_{i}^*(v_{j})=\delta _{ij}, \quad 1\leq i,j\leq n.$.
Then, every $v\in V$ is of the form $v=a_{1}v_{1}+\cdots +a_{n}v_{n}$, which in turn can be represented by a column vector 
$\pmatrix{a_1\\ \vdots\\ a_n}$.
Likewise, every $\varphi \in V^*$ is of the form $\varphi =b_{1}v_{1}^*+\cdots +b_{n}v_{n}^*$ which can be represented by the row vector 
$(b_1,\cdots ,b_n)$
Now, using the definition of the $v^*_i$, it is easy to see that 
$(\varphi ,v)\mapsto b_1a_1+\cdots +b_na_n.$ which is just 
$(b_1,\cdots ,b_n)\pmatrix{a_1\\ \vdots\\ a_n}$