Almost all answers I see mirror this one:

"Suppose we have a mapping φ:Rn→Rm that brings our vectors in Rn to some feature space Rm. Then the dot product of x1 and x2 in this space is φ(x1)Tφ(x2). A kernel is a function k that corresponds to this dot product, i.e. k(x1,x2)=φ(x)Tφ(x2)."

Question 1: I know x1 and x2 are two of our data points (or vectors of features), but why are we getting the dot product of two random data points (out of the hundreds or thousands of data points that we have)?

Question 2: We basically have data points with N number of features. Our data points are linearly inseparable, hence the whole point of the kernel trick is to create an additional feature that offers linear separability (the possibility of separating data with a hyperplane). How is that extra new feature achieved by using the dot product?


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