Mathematical Models

Homework Assignment #5

Due Monday, March 31, 11:00 am

Homework should be turned in at the BEGINNING OF CLASS. Unless otherwise specified, all problem numbers refer to Giordano, Weir and Fox, the required text for the course. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be your own work. At the top of your homework, please list any students you work with or any other resources you found helpful (websites, books, faculty, etc.)

Section 4.1
Problems:   1, 2, 3, 4

Note: For problem #1d, estimate the slope and intercept of your line graphically. For problem #1e, use Least Squares to calculate the best linear fit and compare your answers to part #1d. Use the least squares fit to do the rest of the problems #1f, 2, 3, and 4.

Section 4.2
Problems:   4

Note: For this problem, try dividing the x-data by 10 (or perhaps 100) to get a numerically stable solution. In other words, we don't want the Vandermonde matrix to have entries which are too large, although MAPLE may be able to handle it either way.

Section 4.3
Problems:   1, 3, 4, 7

Section 4.4
Problems:   1a, 1b

Note: For these problems, give the set up (8 linear equations) and then use MAPLE to solve the system.

Additional Problem: Let A_n be the n x n Vandermonde matrix. Show that (x_j - x_i) divides det(A) if i is not equal to j. (In other words, show (x_j - x_i) is a factor of det(A).) Try and use a row operation to help you. Conclude that the product of all such binomials with i < j divides det(A). For extra credit prove that the product of all such binomials with i < j is actually equal to det(A).