Least squares and qr factorization. suppose a is an m n matrix with linearly independent columns and qr factorization a = qr, and b is an m-vector. the vector a^x is the linear combination of the columns of a that is closest to the vector b, i. e., it is the projection of b onto the set of linear combinations of the columns of a. (a) show that a^x = qqt b. (the matrix qqt is called the projection matrix.) (b) show that = bk2. (this is the square of the distance between b and the closest linear combination of the columns of a.)

An automobile assembly line operation has a scheduled mean completion time, μ, of 12 minutes. the standard deviation of completion times is 1.6 minutes. it is claimed that, under new management, the mean completion time has decreased. to test this claim, a random sample of 33 completion times under new management was taken. the sample had a mean of 11.2 minutes. assume that the population is normally distributed. can we support, at the 0.05 level of significance, the claim that the mean completion time has decreased under new management? assume that the standard deviation of completion times has not changed.

A.s.a.! similarity in right triangles, refer to the figure to complete this proportionc/a = a/? a.) cb.) hc.) rd.) s

Which of the functions below could have created this graph?