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Question regarding kernelized ridge regression

Good morning, I was a bit confused by how the expression for $$\alpha^{*}$$ was obtained in the alternative formulation for ridge regression:

Screenshot 2020-12-28 at 09.54.31.jpg

Is this simply done by replacing all instances of $$w^{}$$ with $$X^{T}\alpha^{}$$ and rearranging? I would appreciate an overview on how this is worked out if possible!

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