Meaning of passing from a column to a row vector When passing from column to row vectors in $K^n$ conceptually we're passing from a vector $(a_1,\ldots , a_n) \in K^n$ to it's associated linear functional defined by $ f(x_1,\ldots , x_n)=\sum_i a_i x_i.$ 
Now my question is: What does the inverse operation - passing from a row vector to a column vector in a coordinate space - conceptually mean ?
 A: First, notice that the map
$$\Phi: K^n \to (K^n)^*$$
you've defined by
$$\Phi(a_1, \ldots, a_n)(x_1, \ldots, x_n) := a_1 x_1 + \cdots + a_n x_n$$
is not independent of the choice of basis of $K^n$; that is, if we pick a new basis of $K^n$, in general the map $\Phi$ changes.
Now we do have a canonical basis of $K^n$, namely the one whose $k$th element is the vector with $1$ in the $k$th slot and $0$ elsewhere, and provided we stick with this basis $\Phi$ is well-defined. We can, however, introduce a new object on $K^n$ that simultaneously gives an invariant (i.e., basis-independent) description of this operation and gives a conceptual description for changing between a row and a column vector.
Suppose $K$ is $\mathbb{R}$ or $\mathbb{C}$, and consider the standard inner product on $K^n$ defined by
$$\langle x, y \rangle := x_1 y_1 + \cdots + x_n y_n.$$
Then, for any $a \in K^n$, we have
$$\Phi(a)(x) = \langle a, x \rangle$$
for all $x \in K^n$, so we $\Phi$ (which sends a column vector to a row vector) sends $a$ to the map $K^n \to K$ that sends $x$ to its inner product with $a$. Using a basis it's easy to show that this map is bijective, so the inverse map simply maps such a "pairing" function $K^n \to K$ back to the vector $a \in K^n$ that defines it.
More generally, given an abstract (finite-dimensional) vector space $\mathbb{V}$ over $\mathbb{R}$ or $\mathbb{C}$ together with an inner product $\langle \, \cdot \, , \, \cdot \, \rangle$ on $\mathbb{V}$, we can define a map $\widetilde{\Phi}$ invariantly by
$$\widetilde{\Phi}(a)(x) := \langle a, x \rangle,$$
and by the above this coincides with $\Phi$ in the setting of $K^n$ endowed with the canonical basis and standard inner product. Note that in general this map is not simply writing the entries of a column vector as the entries of a row vector; this is only true in an orthogonal basis. In particular, the canonical basis on $K^n$ is orthogonal for the standard inner product defined above.
Applying such a map $\widetilde{\Phi}$ is a special case of what in the language of tensors is sometimes called lowering an index with the given inner product, and we sometimes denote $a^{\flat} := \widetilde{\Phi}(a)$; likewise, applying $\widetilde{\Phi}^{-1}$ is sometimes called raising an index, and we similarly denote $\alpha^{\sharp} := \widetilde{\Phi}^{-1}(\alpha)$.
