MA 305 Homework 4

Due 11:59 pm on Friday May 2,  2003

Calculations necessary for these problems may be done either by hand or with Maple. Solutions are to be submitted as a ASCII text (.txt) or Maple Worksheet (.mws) (version 7 or lower) file via WolfWare.

If you use Microsoft word, please save your file as ascii text with a .txt extension before submitting the file.

Incorrectly submitted files will result in NO CREDIT, so be sure the file you submit is readable in a text editor or Maple. If you are using Windows, check text files using notepad to make sure it looks readable.

Show your work for full credit. This means show all of your maple commands or if done by hand show most of the intermediate steps so we know how you arrived at your answer.

You must write up your own homework. You may work together in the formulation of your answers, but you must type your own file(i.e. no copying and pasting either).

Put your name and the names of all the people you workedwith at the beginning of your submission file.

Questions about the assignment may be sent to the TA, Lauren D'Elia at rldelia@math.ncsu.edu.

1. Let
[   1 ] [   1 ] [   1 ]
[   1 ]   [ -1 ] [   1 ]
u = [   1 ]  , v = [  2 ]  and  w = [   4 ]
[   1 ] [ -2 ] [   4 ]
[   1 ] [   3 ] [   9 ]

Compute the following

  1. The scalar products: uTv, uTw, vTw.
  2. The Euclidean norm: ||u||, ||v||, ||u+v||.
  3. The angle between v and w.
  4. The projection of v onto u.
  5. A basis for the orthogonal complement of V=spanR(u,v,w).
2. Let
[ -3 ]
[ -2 ]
b = [   6 ]  .
[ -4 ]
[ -8 ]
  1. Compute b^ , the orthogonal projection of b onto V = spanR(u,v,w) from Problem 1. Express b^ as a linear combination of u,v,w.
  2. Verify that b - b^ is orthogonal to u,v,w. Express b - b^ as a linear combination of the basis vectors for the orthogonal complement that you have found in Problem 1.
  3. Compute the Euclidean, infinity, and 1-norm of b - b^.

3.  Consider the list of x,y coordinates of points.
x y
 1 -3
-1 -2
  2   6
-2 -4
  3 -8
  1. Problem 2 is a least squares fit for the quadratic function y = a + bx + cx2 for the date values. Which values for a,b,c are found in Problem 2?
  2. Do a least squares fit on the same list of points for the alternative model y = a + bx + cx3. In terms of Euclidean norm, which of the two models comes closer to the values of y, i.e., has a smaller sum-of-squares-of-differences residue.
4. Let
[   1 ] [ -5 ] [   4 ] [ -3 ] [ -5 ]
[   0 ] [   6 ] [ -7 ] [ 11 ] [ 19 ]
u = [ -1 ]  ,  v = [   5 ]  ,  w = [  -4 ] , z = [   1 ]  and  b = [ 3 ]
[   0 ] [   6 ] [   1 ] [ -5 ] [ -13 ]
[   3 ] [   7 ] [   1 ] [   5 ] [ -1 ]
  1. Using the Gram-Schmidt process, convert u,v,w,z into an orthogonal basis for the spanR(u,v,w,z).
  2. Let A = [u v w z] (the columns of A are the vectors u,v,w,z). Using results from part a), determine the QR factorization of A.
  3. Use the QR factorization from part b) to solve the linear system Ax = b in x.