4. (20 pts) Suppose you have n vectors 21, --- , In in RM. Consider a line L going through some point to ERM and with some orientation WER" and ||0|| = 1 (the equation of this line is 10 +tw where t is a scalar). Now consider the line that is closest to the point in the sense that it minimizes distance(li, L) i=1 (the sum of squares of the distances between the vectors {1}}}=1 and the line). 1) (10 pts) Show that the slope of the closest line is the first principal component. 2) (10 pts) Show that this line goes through the average vector =- 2 n i=1 4. (20 pts) Suppose you have n vectors 21, --- , In in RM. Consider a line L going through some point to ERM and with some orientation WER" and ||0|| = 1 (the equation of this line is 10 +tw where t is a scalar). Now consider the line that is closest to the point in the sense that it minimizes distance(li, L) i=1 (the sum of squares of the distances between the vectors {1}}}=1 and the line). 1) (10 pts) Show that the slope of the closest line is the first principal component. 2) (10 pts) Show that this line goes through the average vector =- 2 n i=1