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When dealing with Linear regressin with dependent variables, one can consider the optimization problem: $$\arg \min_w 0.5\lVert{w}\rVert^2 \\ s.t. Xw=y$$

Where $X\in\mathbb{R}^{n,d}$ is the data matrix and $y\in\mathbb{R}^{n}$ is the labels vector.
Using Lagrangian multipliers I was able to show that $w^*$ can be written as $w^*=X^T\alpha$ for $\alpha \in \mathbb{R}^n$.

Now given a new $x\in\mathbb{R}^d$, consider $x^Tw^*$. How can this be expressed using dot products between $x\in\mathbb{R}^d$? The goal is to show that the kernel trick is working for that setup of linear regression.

Thanks.

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