 Solution

LDA assumes $Y\in \{1,\dots ,L\}$ with prior probabilities ${\pi }_l>0$ andwhere the class means may differ but the nonsingular covariance matrix is common. Bayes' rule chooses the class maximizing $\log {\pi }_l+\log f_l(x)$. Cancelling terms common to every class gives the discriminantThe boundary between classes $l$ and $m$ is ${\delta }_l(x)={\delta }_m(x)$, an affine hyperplane with normal ${\Sigma }^{-1}({\mu }_l-{\mu }_m)$. Hence the Bayes rule $\arg {\max }_l{\delta }_l(x)$ is a linear classifier.