In lecture, we derived Linear Discriminant Analysis $($LDA$)$ by starting with the Bayes classifier and modeling

each class$-$conditional density as a multivariate Gaussian and using the same covariance matrix for each. We

stated, but did not prove, that the decision boundary of an LDA classifier is linear.

Recall that, for a binary classifier based on the Bayes Classifier, the decision boundary is the set of all points

vec$(x)$ where

hat$(p)(vec(x)|Y=1)hat(P)(Y=1)=$hat$(p)(vec(x)|Y=0)hat(P)(Y=0),$

where the various hat$(p)$ and hat$(P)$ are estimated densities and probabilities.

Using this fact, prove that the decision boundary of an LDA classifier is linear. For simplicity, you may

assume that vec$(x)in{R}^2$ and that the shared covariance matrix is diagonal $($although the result holds even if the

covariance matrix is not diagonal$).$

Hint: since you may assume that vec$(x)=(x_1,x_2{)}^T,$ you can start from the above equality and solve for $x_2$ in

terms of $x_1,$ showing that you get the equation of a line.