Problem 1 Let y = (1,-1) and u = (-1,3). NOTE these are vectors. (a) Compute the orthogonal projection of y onto u. (b) Find the distance from y to the line passing through 0 and u. Problem 2 Let y = (-1,4,3) , u1(1,1,1) u2 (-1,3,-2). NOTE: these are vectors Find the closest point to y in the subspace W = span {u1, u2} Solution a).

1. Problem 2.9 please Suppose that men and women are distributed in the freshman and soph
degree nl(>r college according to the proportions listed in the following table. A student is
chosen at random and let Af, V,F. and S be the events, respectiel>, tl ^ student is a man, a
woman, a freshman, or a sophomore. Then, being a man or a worn a i being a freshman or a
sophomore are independent, if: Determine flie number x so that the preceding independence
relations hold. Hint. Determine x by using any one of the above four relations (and check that
this value of x aiso satisfies the remaining three relations). The r.v. A' has p.d.f. given by:
Solution
P(M Intersection F)=4/16+x
P(M).P(F)=100/(16+x)^2
so P(M Intersection F)=P(M).P(F) implies 16+x=25 so x=9