Design Matrix

The design matrix $\Phi$ is an $N\times M$ matrix where each row contains the basis function values for one data point:

$$\Phi =\left(\begin{array}{c}\varphi (x_1{)}^T\\ \varphi (x_2{)}^T\\ ⋮\\ \varphi (x_N{)}^T\end{array}\right)=\left(\begin{array}{cccc}{\varphi }_0(x_1)& {\varphi }_1(x_1)& \cdots & {\varphi }_{M-1}(x_1)\\ {\varphi }_0(x_2)& {\varphi }_1(x_2)& \cdots & {\varphi }_{M-1}(x_2)\\ ⋮& ⋮& \ddots & ⋮\\ {\varphi }_0(x_N)& {\varphi }_1(x_N)& \cdots & {\varphi }_{M-1}(x_N)\end{array}\right)$$

This allows us to write all predictions compactly as $\Phi w$ instead of computing $w^T\varphi (x_n)$ for each point individually.