Exercise $1$: Feature Map

Let $x,yin{R}^2,$ i$.$e$.x=(x_1,x_2)$ and $y=(y_1,y_2).$ Define polynomial kernel $K$:$R^2{R}^{2}R$ by $K(x,y)=(x^{TT}y{)}^2.$ Find the feature map $$:$R^2{R}^D.$ Please specify the dimension $D.$

Consider the same kernel function $K(x,y)=(x^{TT}y{)}^2$ which is defined on $R^3{R}^3$ in this part, i$.$e$.x,yin{R}^3.$ What is the feature map $$:$R^3{R}^D?$ What is the dimension $D?$

In general, suppose that $x,yin{R}^d$ and polynomial kernel is defined as $K(x,y)=(x^{TT}y{)}^2.$ Without writing the feature map $$:$R^d{R}^D$ explicitly, write down the dimension $D$ as a function of input dimension $d.$