Most texts I've seen hide a lot of what's under the hood in SVMs (support vector machines). The book I'm reading says SVMs can be solved using quadratic programming by using slack variables, giving solution: $$w^* = \sum _{i=1}^N \alpha _i \boldsymbol x_i$$ where $\boldsymbol x_i$ are the training samples and $\alpha _i$ are the dual variables. There's an analogy here to the kernel trick applied to ridge regression where in the ridge regression case, $$\boldsymbol \alpha = (K + \lambda I)^{-1} \boldsymbol y$$
In other words, $\alpha_i$ depends on the Gram Matrix $K$ which depends on the kernel function.
Question 1: in the SVM case do the dual variables $\alpha_i$ also depend on the kernel function? If not, why? If so, how is this implemented/dealt with in finding the solutions? My concern here is the kernel trick for ridge regression only worked because we could write down $\boldsymbol \alpha$ in terms of $X X^T \rightarrow K$ via kernel trick. But if we're solving $\boldsymbol \alpha$ using quadratic programming, it's not clear how we can get the $X$ terms in the form $XX^T$ to apply the kernel trick (and hence avoid transforming the data features explicitly)?
Question 2: do most SVM packages find the solution in the dual space ($\alpha$'s) rather than primal space ($w$'s)?
