Question $3($Kernel methods$)(35$ pts$)$:$.$ Answer the following questions:

$($a$)$ Suppose $K=([9,4],[4,1]),$ find the centered kernel matrix.

$($b$)$ Suppose $K=([4,7,1,5],[7,2,6,3],[1,6,9,8],[5,3,8,5]),$ find the distance between $(x_2)$ and $(x_4)$

$($c$)$ Given three points $x_1=(2,3{)}^T,x_2=(3,4{)}^T,x_3=(2,4{)}^T.$

Find the kernel matrix using the Gaussian kernel assuming that ${}^2=5$

$($d$)$If the number of dimensions of the original space is $5,$ what is the number of dimensions of the feature space for the Gaussian kernel?