Q$1.(25$ marks$)$ Consider a regression problem where the input vector is two$-$dimensional.

The training set consists of four examples with the feature vectors x$(1)=(0,0)$T $,$ x$(2)=$

$(2,10+2$D$2)$T $,$ x$(3)=(0,5+$ D$2)$T $,$ x$(4)=(2,5+$ D$2)$T and the targets t$(1)=20$ D$4,$

t$(2)=20+3$D$4,$ t$(3)=$ t$(4)=20+$ D$4.$ The test set consists of one example with feature

vector x$(5)=(2,8+2$D$2)$T and target t$(5)=20+3$D$4.$

You decide to first reduce the dimension of the feature vector to $1$ by using principal

component analysis. Thus, PCA is applied on the set of the feature vectors correspond$-$

ing to the four examples. After that you train a linear regression model with the new

one$-$dimensional feature as input to obtain your predictor. Let $($x$)$ in R denote the new

one$-$dimensional feature corresponding to the input vector x $($$($x$)$ is obtained using

PCA$).$ Thus, your predictor has the form f $($$($x$))=$ w$0+$ w$1$$($x$).$

Find the vector of weights obtained after training. Show the formula of the predictor in

terms of the input vector x$.$ Compute the training error and the test error. Show your

work.