Hi Slutsky,
To answer your query, the various notations you come across in the kernel literature is dependent on the user group. From a notation perspective x, y (or z) refer to vectors where as xi and xj refers to element i and j within vector x.
A kernel matrix is usually donated with capital K whereas k is the kernel function (dot product). Hence, where as K is a matrix k(x,y) will be only an entry scalar in matrix K.
To answer your second question regarding dependence and independence. The assumption is that the data is iid (identically and identically distributed).
Perhaps this pseudocode will help understand the notation
X - matrix of size nxm (n sample and m features)
K - matrix of size nxn
for i=1 to n
for j=1 to n
K[i,i] = X[i,:]*X[j,:] % a linear dot product
endfor
endfor
a good book is 'introduction to support vector machine' by cristinanini and shawe-taylor
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