Linear Algebra Math 310
Spring 2017
Assignment Sheet 3

This sheet specifies a selection of exercises from the text, and in some cases a few additional exercises. The notation 1-21o means the odd problems between 1 and 21, inclusive.


Section 5.3   Regular: 1, 5, 8, 9, 13, 15.    
The following additional problems are regular.  For all of these problems, let the matrix A be given by

A = 0 1 1 1 matrix
  1. Diagonalize A
  2. Use the diagonalization of A to find a formula for Ak
  3. Use your diagonalization of A to find a formula for Akx where x = [x  y]T  and x and y can be anything.
  4. Use your result from the previous problem to find a formula for Akx where x = [1  1]T
  5. Compare the formula in part 4 to the formula you found in the extra exercises for 5.2, above
  6. Review the example on pages 301 and 302 at the start of chapter 5.  Explain how diagonalizing the matrix in that example would contribute to understanding what the model predicts about the future for the spotted owl population.


Section 6.1   Regular: 1, 3,  5, 7, 11, 13, 15, 18, 19.  Optional Problems: 22*, 27*, 29*. 


Section 6.2    Regular: 1-23o, 24, 27-29,.


Section 6.5  Instead of reading the text, read this handout.    Then do the following problems:  Regular: 1, 3, 5, 7, 11, 17, Optional, using freemat: 25. 


Section 6.6.  In addition to reading the text, you may find it useful to review this outline from class.   1, 5, 6, 7, 8, 15, 16 (consider using Cramer's rule).

The following additional problems are regular.

  1. Find a polynomial ax2+bx+c that comes as close as possible to interpolating the points (0,1), (1,2), (2,4), (3,8)
  2. One year, the average monthly temperatures in Green Bay, Wisconsin,were as shown in the table below:

Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Avg. Temp

15.4

18.0

28.6

43.8

54.5

64.5

69.2

67.7

58.9

49.2

34.1

20.9

It is natural to suppose that these average temperatures follow a sine shaped curve reflecting the seasonal variation over the year. If t is the time in months, then we expect that the sine will repeat after t = 12, so the period is 12. With that in mind, the form of the sine curve should be

a sin 2 pi t /12 + b cos 2 pi t / 12.

Find the constants a, b, and c which make the resulting sine curve come as close as possible to the actual data.