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

- Diagonalize
*A* - Use the diagonalization of
*A*to find a formula for*A*^{k} - Use your diagonalization of
*A*to find a formula for*A*^{k}**x**where**x**= [*x y*]^{T}and*x*and*y*can be anything.

- Use your result from the previous problem to find a formula for
*A*^{k}**x**where**x**= [1 1]^{T}

- Compare the formula in part 4 to the formula you found in the extra exercises for 5.2, above
- 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.

- Find a polynomial
*ax*^{2}+*bx*+*c*that comes as close as possible to interpolating the points (0,1), (1,2), (2,4), (3,8) - 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

.

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