MA 405 Homework 4

Due 4:59 pm on Friday April 24,  2009

This assignment will be turned in on paper.   Either hand in your  assignment  in class on Wednesday,  April 22,  or place it in  Kaltofen's  mailbox in HA 245  before 5pm on  Friday, April 24.

If you use Maple to solve some of the questions, then simply print your worksheet out and attach to it the problems that you did by hand.

Please clean up your Maple worksheets and clearly label the problems. Remember you can omit output on commands by using a colon e.g. with(LinearAlgebra):


1.  Consider the list of x,y,z coordinates of points.
x y z
 1 -3 4
-1 -2 2
  2   6 11
-2 -4 7
  3 -8 22
  1. Problem 1 is a least squares fit for the quadratic function z = ax + by + cxy for the data values. Which values for a,b,c are found?
  2. Please do a least squares fit on the same list of points for the alternative model z = ax2 + by2 + c. In terms of Euclidean norm, which of the two models comes closer to the values of z, i.e., has a smaller sum-of-squares-of-differences residue.
2. Let


[   0
 ]


[
   3
 ]


[   1
 ]


[   2
 ]


[    6
 ]


[  -1
 ]


[   2
 ]


[  -2
 ]


[   1
 ]


[    4
 ]
u = [   0
 ]  ,  v = [   3
 ]  ,  w = [  -3 ] , z = [   0
 ]  and  b = [    6
 ]


[   1
 ]


[  -2
 ]


[   2
 ]


[   1
 ]


[    0
 ]


[  -2
 ]


[  -5
 ]


[  -1
 ]


[  -3
 ]


[  -14
 ]
  1. Using the Gram-Schmidt process, convert u,v,w,z into an orthogonal basis for the spanR(u,v,w,z).  Please do not normalize the  orthogonal vectors to unit length.
  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.
3.  Consider the following ``weighted infinity norm'' function on R2: F([x,y]T) = max{2|x|,|y|/3}.  Please prove that F is a norm function, that is, F satisfies the conditions (N1)-(N3) from class.

4.  Determine if the given functions from R3 to the given spaces are linear transformations. If it is a linear transformation, find the matrix A such that L(v)=A*v. If it is not a linear transformation, show why not.
    1. L([x,y,z]T)=[-x+5z, 2y]T in R2
    2. L([x,y,z]T)=[x+1, y+2, z+3]T in R3
    3. L([x,y,z]T)=[z-y, y-x, x-z]T in R3