Transformation from one feature space to another I have found the following example:


  As an example consider the case when the input space  $ {\mathcal{X}}$
  consists of images of  $ 16\times 16$ pixels, i.e. $ 256$ dimensional
  vectors, and we choose $ 5$th order monomials as non-linear features.
  The dimensionality of such space would be
  
  $\displaystyle \left(\begin{array}{c}5+256-1\\ 5\end{array}\right) \approx 10^{10} \;$     
  
  Such a mapping would clearly be intractable to carry out explicitly.
  We are not only facing the technical problem of storing the data and
  doing the computations, but we are also introducing problems due to
  the fact that we are now working in an extremely sparse sampled space.
  By the use of the SRM principle, we can be less sensitive to the
  dimensionality of the space and achieve good generalization.
  
  The problems concerning the storage and the manipulation of the high
  dimensional data can be avoided. It turns out that for a certain class
  of mappings we are well able to compute scalar products in this new
  space even if it is extremely high dimensional. Simplifying the above
  example of computing all $ 5$th order products of $ 256$ pixels to
  that of computing all $ 2$nd order products of two ''pixels'', i.e.
  
  $\displaystyle {\boldsymbol{x}} = (x_1,x_2)$   and$\displaystyle \quad\Phi({\boldsymbol{x}}) = \left(x_1^2,\sqrt{2}x_1x_2,x_2^2\right)\;$


The problem that I have is in understanding the last formula, from where does the author gets those values for the individual transformation? To me, it seems that he is applying the formula of analytical geometry of a circle, but I am not quite sure.

Also how does the author apply the transformation from $ {\mathcal{X}}$ to $\displaystyle\quad\Phi({\boldsymbol{x}})$?
This is just the explicit form of the feature map for a homogenous, degree 2 polynomial kernel.  Here your input space is $\mathbb{R}^2$ and the feature space can be identified with $\mathbb{R}^3$.  The point the author is making is that instead of computing the feature map explicitly, and then taking the Euclidean inner product $\langle \Phi(p),\Phi(q)\rangle$, you can just compute $\langle p,q\rangle^2$.