M346, Applied Linear Algebra

This page is always under construction, so
you
should check it regularly. Assignments with the **due date**
in **bold face** are set in stone.
Other assignments are still tentative.

Starred problems have solutions in the back of the book. You do **not**
have to turn them in, but you may need to look at them in order to
understand the subsequent problems.

Section 2.3, Page 23 # 7, 9

Exploration on page 24, steps 1-3. For step 2, there is a typo, and the polynomial should be

Section 3.1, page 42, #6

Exploration on page 43: Steps 1-10 (you can skip 11) Feel free to use MATLAB (or any other piece of technology) to actually do matrix products. Some of the steps explain the questions of the earlier steps (e.g., step 6 gives away the answer to step 5). For your own learning, please work all the steps in order.

Section 3.3, Page 50 #1*, 2, 6. On problem 2 you can use the result from problem 1, but you'll need to compute P

Section 3.5, Page 56, #4, 7

Note that you are not required to turn in the starred problems.

Section 4.2, Page 61, #2, 3, 5*

Section 4.4, Page 71, #3*, 6, 9.

Section 4.6, Page 81, #3, 6, 7, 9

Section 4.9, page 94, #16*, 17*, 19*. Note that all of these are starred, so you don't actually have to turn in anything from this section! Still, working these problems and comparing your answers to the solutions is a VERY good way to understand power vectors.

Exploration after section 5.1, steps 1, 3, 4. Steps 5-8 are also interesting -- give them a look if you have the time (but don't turn them in).

Section 5.3, page 115, #1*, 2, 5

Exploration on pages 119-120, steps 1-5 (the whole exploration)

Section 5.5, page 125, #2, 5

If you have the time, look at the exploration on page 143 (but don't turn it in). Getting a good feel for this system will go a long way towards understanding how linearization works. This system of equations describes two species that compete fiercely, more fiercely than the system we studied in class. In the exploration's example, coexistence is unstable, and the only stable equilibria are where one species has driven the other to extinction.

Section 6.2, page 151, #4, 5*, 7

Section 6.4, page 161, #1*, 2, 5

Section 6.5, page 166 #1

Section 6.7, page 175, #2*, 3, 4

Exploration after 6.9 (page 188), parts 1 and 5. (The whole exploration is recommended, but only turn in parts 1 and 5.)

Incidentally, the census bureau announced the 2010 population of Austin in March. Their number for the city of Austin proper is 790,390. This is probably less that you are "predicting" with your exponential least-square plots. However, this has more to do with politics than with population trends. As the city of Austin has grown over the last 162 years, the city limits have repeatedly expanded.

Greater Austin, with a population of over 1.7 million, is growing as fast as ever. However, there has been almost no annexation of land to Austin in the last decade. Most of the recent growth has been outside the city limits, and those communities (e.g. Pflugerville and Round Rock) are likely to remain independent.

Here is how the Austin Metropolitan Statistical Area (MSA) has grown since 1970. Back in 1970, the city of Austin was about 63% of the MSA. In 2000 it was about 52%. Now it's down to 46%.

1970: 398,938

1975: 497,400

1980- 585,051

1985- 758,510

1990- 846,227

1995- 1,031,557

2000- 1,249,763

2005- 1,452,529

Section 7.4, page 213, #4, 5, 6 (Find the eigenvalues exactly, and by hand, but you don't need to find the eigenvectors.)

Section 7.5, page 219, #1, 2. I did part of problem 2 at the board today, stating the answers and showing that v was an eigenvector of K

Also, if you are a physics student, or are otherwise interested in quantum mechanics, I

Section 8.4 (page 242), #2*, 3, 4

Problems 5-7 on page 242 show how a continuous medium can be approximated by a discrete one. This is important both for physics (a string is actually a finite collection of atoms) and for computer science (solving differential equations numerically with a finite number of terms). These problems are

**Practice exercises**

These exercises, covering material after the last homework,
are to help you get ready for the final. They are **not** to be turned in.
To give you feedback,
I've tried to assign as many starred problems, with solutions in the
book, as possible.

Section 8.4 (page 242), #2*, 3, 4, 5, 6*, 7

Section 8.5 (pages 249-250), #3*, 5*, 6*, 13*, 16*, 17