# Show Unbiased Learner

I want to show that $$g_{\tau}(\mathbf{x}) = \mathbf{x}^T\hat{\beta}$$ where $$\hat{\beta} = \mathbf{X}^+\mathbf{y}$$ and $$\tau$$ denotes a fixed training set is an unbiased learner, in the sense that: $$\newcommand{\Tau}{\mathrm{T}}E g_\Tau(\mathbf{x}) = g^*(\mathbf{x})$$

where $$g^*(\mathbf{x})=\mathbf{x}^T\beta$$ is the optimal prediction function, where T denotes a random training set.

I think it's possible to use the Tower property: $$EE(X|Y)=E(X)$$ but I'm not sure how to proceed.