MA 305 Homework 5

Due 11:59 pm on Friday April 26,  2002

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.

Submitted files should have either .txt or .mws extensions. This makes the grader's life much easier.

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 worked with at the beginning of your submission file.

Questions about the assignment may be sent to the TA, George Yuhasz at  gyuhasz@math.ncsu.edu


1.  Let the following be a list of points in R3.
x y z
 0  0 6.98
 1 -1 18.01
 1 1 10.01
-1  1 3.97
-1 -1 8.04
 1  2 6.02
-2  3 7.98
 2  3 12.03
 2  4 7.04
 2  2 16.99

Let us guess that these points all lie on the plane z = a + b*x + c*y. We want to do a Least Squares fit to solve for the coefficients a,b and c of the plane.

  1. Find a matrix A and a vector b such that if x=[a,b,c]T then a solution to Ax=b would yield the coefficients of the plane.
  2. Do a Least squares fit to find x*and b* such that Ax*=b*.
  3. How close is the least squares fit (i.e. what is the euclidean norm of b-b*).
2. What if the points from problem 1 lie on a different type of surface? Let's guess they lie on a quadratic surface whose equation looks like z = a + b *x + c*y + d*x*y + e*x2. Repeat steps a-c of problem one, and this time let x=[a,b,c,d,e]T. Which of the two fits is better, the plane or the quadratic surface? Why?

3. Let the following vectors be in R3.
 
[ x1 ] [ y1 ]
x = [ x2 ]  and  y = [ y2 ]
[ x3 ] [ y3 ]

Determine if the following are inner products of R3. Explain why or why not.

  1. <x,y>=(x1)2*y1+x2*y2+x3*y3
  2. <x,y>=x12+y22+x3*y3
  3. <x,y>=|x1|+|y1|+|x2|+|y2|+|x3|+|y3|
  4. <x,y>=2*x1*y1 - .00001*x2*y2 + 1000000*x3*y3
  5. <x,y>=2*x1*y1 + .00001*x2*y2 + 1000000*x3*y3
4. Let
[   1 ] [ -1 ] [   4 ] [   6 ] [   5 ]
[   0 ] [   0 ] [ -3 ] [ -3 ] [   7 ]
u = [ -1 ]  ,  v = [   2 ]  ,  w = [   1 ] , z = [ -2 ]  and  b = [ -4 ]
[   0 ] [ -1 ] [   0 ] [   1 ] [   3 ]
[   3 ] [   0 ] [   0 ] [   3 ] [ -2 ]
  1. Using the Gram-Schmidt process, compute an orthogonal basis for the spanR(u,v,w,z).
  2. Use the basis from part a to compute the orthogonal projection of b onto the spanR(u,v,w,z).