Question $1$: Consider a linear model of the form hat$(y{)}_i=$hat$(f)(x_i)=w_o+w_1{x}_i+w_2{x}_i^2+cdots{x}_i^{M-1}{w}_{M-1}=w^T{x}_i,$ where $w,x_{i}in{R}^M$

with:

$w=\left[\begin{array}{c}{w}_o\\ w_1\\ \text{c}\text{d}\text{o}\text{t}\text{s}\\ w_{M-1}\end{array}\right],x_i=\left[\begin{array}{c}{x}_i\\ x_i^2\\ \text{c}\text{d}\text{o}\text{t}\text{s}\\ x_i^{M-1}\end{array}\right]$

Next, we are given a training dataset having $N$ examples as$,D=\{(x_i,y_i)\}$ for $i=1,2,$cdots,$N$ denoting the corre$-$

sponding input$-$target example pairs. For this question, use the following loss function:

$L(w)=\frac{1}{2}{\displaystyle \underset{i=1}{\overset{N}{}}}(hat(y{)}_i-y_i{)}^2+\frac{1}{2}{\displaystyle \underset{i=1}{\overset{N}{}}}{y}_i^2$

a$.$ Assume $=0,$ derive a closed form expression for finding the values of weight vector $w.$ Write down proper size

and structures of every vector and matrix used.