A "linear separator" separates points using a straight line $($in one$-$dimensional space$),$ a plane $($in two$-$dimensional space$),$ or a hyperplane in higher dimensions. We built a one$-$dimensional linear separator in class to classify wine as red or white.

a$.$ Show graphically that the VC dimension of a linear separator with $1$ dimension is exactly $2.$

b$.$ Show graphically that the VC dimension of a linear separator with $2$ dimensions is exactly $3.$

c$.$ What is the VC dimension of a linear separator in d dimensions? Prove it$.$