MA 405 Homework 4


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