Show that [mathbf{w}^{top} mathbf{X}^{top} mathbf{X} mathbf{w}=w_{0}^{2}left(sum_{n=1}^{N} x_{n 1}^{2} ight)+2 w_{0} w_{1}left(sum_{n=1}^{N} x_{n 1} x_{n 2} ight)+w_{1}^{2}left(sum_{n=1}^{N} x_{n

Show that

\[\mathbf{w}^{\top} \mathbf{X}^{\top} \mathbf{X} \mathbf{w}=w_{0}^{2}\left(\sum_{n=1}^{N} x_{n 1}^{2}\right)+2 w_{0} w_{1}\left(\sum_{n=1}^{N} x_{n 1} x_{n 2}\right)+w_{1}^{2}\left(\sum_{n=1}^{N} x_{n 2}^{2}\right)\]

where

\[\mathbf{w}=\left[\begin{array}{l} w_{0} \\ w_{1} \end{array}\right], \mathbf{X}=\left[\begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \\ x_{31} & x_{32} \\ \vdots & \vdots \\ x_{N 1} & x_{N 2} \end{array}\right]\]

(it's probably easiest to do the \(\mathbf{X}^{\top} \mathbf{X}\) first!)