Linearly Independent Sets in Vector Spaces induced by Kernels • /r/MachineLearning

I hope this post is okay (if not let me know). I'm attaching a pdf which rigorously defines my question. Briefly, what I'm wondering is this - for the set of data points {x1,...,xp} in a vector space, (say, Rn) under what conditions is the set {k(x1,),...,k(xp,)} (where k(,) is a kernel function) independent? What conditions must the set {x1,...,xp} and the kernel function have to ensure independence? If there isn't an immediate answer to this question I'll happily take recommendations for mathematical reading towards trying to answer this question.