$1\mathrm{.}8$ Linear Discriminant Analysis

Consider the following $7$ observations of two classes with $2$ measured variables

each:

$\left[\begin{array}{ccccccc}2& 1& 1& 0& -1& -1& -2\\ 2& 1& 3& 2& 3& 1& 3\\ c1& c1& c1& c1& c2& c2& c2\end{array}\right]$

where each row represents one variable $($last denoting their class belonging$)$ and

each column represents an observation. Our aim is to use Linear Discriminant

Analysis to be able to classify new observations whose class is unknown to us$.$

a$)$ For each class: calculate the corresponding covariance matrix $S_1$ and $S_2.$

b$)$ Use LDA to find the line $w^{**}$ which when projected upon best separates

the two classes. Plot the points and the line $w^{**}($hand$-$drawn plots are

accepted as long as they are accurate$).$