SVM Kernel confusion Suppose that we have an array of 10x2 elements (features). Each of these features are two-dimensional. Something like this:
Array A:

0.001  0.56
0.045  0.12
0.546  0.54
0.123  0.12
1.435  0.01
1.234  0.01
.
.
.
1.654  0.12

Now, if I want to project this feature space in a higher dimension I need to implement a Kernel. Consider the linear kernel: $K(x,y) = (x' \times y +1)$. 
I am confused about this $x$ and $y$. What are my $x$ and $y$ here? Do I need more data? These features are also supposed to belong in two different categories. Do I have to split array A into two different arrays each one contains samples from class A and B, and then do the dot products? 
 A: Let me explain by a simple example with 1D data $x$.
We want to classify + and -, which is quite simple for linear separable data:  

Perfect, we are done, we have the purple decision boundary. But what to do with the following data?

You can not find one straight line separating these data set. So we have to find a mapping $\Phi$ from one dimensional $x$ to a two dimensional feature, let's call it $z$: 
$\Phi(x) \rightarrow z, \quad z = \begin{pmatrix}x & x^2\end{pmatrix}^\top.$ 
Et voilà, we have no a feature space which is again linear separable, but no longer in 1D, but in 2D:

So you can see, there are no further data (features) involved. You do not need find further features (maybe you have to do, but not for the kernel trick), but find a mapping in higher space (in your case, if I got you right, from 2D two maybe 3D or more). And notice it is also possible to have a mapping from 2D to 2D, but transformed. This is one reason why finding and applying the right kernel is not trivial!
