$1.$ Consider the following supervised dataset consisting of $10$ observations, each with two features $($observable data is in X $)$ and an associated label in Y :

X$=[[01$; $00$; $11$; $00$; $11$; $10$; $10$; $11$; $20$; $21]]$ Y$=[[1$; $1$; $1$; $1$; $1$; $0$; $0$; $0$; $0$; $0]]$

$($a$)$ Compute the average weighted entropy of the class label in the subsets created by splitting the dataset based on the value of the first feature. You may assume that the features are categoricial $(5$pts$).($b$)$ Now make the same computation, but as if we created subsets using the second feature! $(5$pts$).($c$)$ Which feature is more discriminating based on results in Part $($a$)?$ That is$,$ which feature provides better class separation $(2$pts$)?($d$)$ What are the principle components of the observed data X $?$ For this $($and the next$)$ part you may assume that the features are continuous and therefore should zscore them. Make sure your final principle components are all unit length. You MAY use a utility function like e i g or s v d to determine these $(5$pts$).($e$)$ In your own words, describe these axis in terms of a conventional $2$D Cartesian Coordinate system $(3$pts$).($f$)$ If we were to project our data down to $1-$D using the principle component, what would the new data matrix X be $(5$pts$).$