# Linear regression with dependent variables: express prediction with dot products

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.