Homework Assignments for
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.

Monday July 11 First day. No homework due.

Tuesday July 12: Chapter 1, Page 7, #1, 2, 5.

Wednesday July 13: Section 2.2, page 20, #2, 4, 5, 8, 12

Thursday July 14: Section 2.1, Page 13, #2, 4, 6, 8, 10.

Friday July 15: Note the last-minute changes!!
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 p(t)=a +bt + ct2 + dt3, not p(t)=a +bt + bt2 + ct3. On step 3, use the points t=0, t=&pi/3 and t=&pi, not the points 0, &pi/2 and &pi . Note that cos(&pi/3)=1/2 and cos2(&pi/6)=3/4.

Monday July 18: Page 28 #6, 8, 12
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.

Tuesday July 19: Section 3.2, #1*, 5, 11. (Note that 5 refers to the result of 4, but you're not required to actually do problem 4)
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 PBE yourself.
Section 3.5, Page 56, #4, 7
Note that you are not required to turn in the starred problems.

Wednesday July 20: Section 4.1, Page 59, #1, 2*, 3, 4*, 6
Section 4.2, Page 61, #2, 3, 5*

Thursday July 21: Section 4.3, Page 67, #1, 4, 6*, 9, 10*
Section 4.4, Page 71, #3*, 6, 9.

Friday July 22: Section 4.5, Page 76, #1, 2, 4, 6
Section 4.6, Page 81, #3, 6, 7, 9

Monday July 25: First midterm. No homework due.

Tuesday July 26: Section 4.8, Page 89, #1*, 5, 6, 8, 9. On problem 6, your final answers should be a matrix that stretches and rotates. By what factor does it stretch? By what angle does it rotate? We already did #8 in class, so writing this up should be easy. I'm including it mostly for the contrast with #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.

Wednesday July 27: Section 5.1, Page 102, #2, 4, 8
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).

Thursday July 28: Section 5.2, Page 107, #2, 3, 4, 7.
Section 5.3, page 115, #1*, 2, 5

Friday July 29: Section 5.4, page 119, #4 (just one problem)
Exploration on pages 119-120, steps 1-5 (the whole exploration)
Section 5.5, page 125, #2, 5

Monday August 1: Section 5.7, page 142, #2, 3, 6, 8, 13
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.
Tuesday August 2: Section 6.1, page 149, #7, 8
Section 6.2, page 151, #4, 5*, 7

Wednesday August 3: Section 6.3, page 156, #2, 4, 6*, 7
Section 6.4, page 161, #1*, 2, 5
Section 6.5, page 166 #1

Thursday August 4: Section 6.5, page 166, #7, 8, 10, 11
Section 6.7, page 175, #2*, 3, 4

Friday August 5: Second midterm. No homework due.

Monday August 8: Exploration on pages 176-7, all 5 parts. Section 6.9 Page 186, #1*, 3
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

Tuesday August 9: Section 7.1 Page 194, #2*, 3, 4, 5*, 6, 7. (For 7, try integrating by parts. This space comes up in quantum mechanical treatments of the harmonic oscillator.)

Wednesday August 10: Section 7.2 Page 199, #1, 3, 4
Section 7.4, page 213, #4, 5, 6 (Find the eigenvalues exactly, and by hand, but you don't need to find the eigenvectors.)

Thursday August 11: Section 7.4, page 213 9, 10. (For 9 and 10, unlike last night's HW, you DO 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 Kv with eigenvalue 0. Finding the other eigenvalues requires additional work. A useful trick is looking at the trace of Kv2.
Also, if you are a physics student, or are otherwise interested in quantum mechanics, I strongly recommend working problems 3* and 4, and the exploration on pages 220-221. This is what angular momentum in quantum mechanics is all about. However, the exploration is not required and will not be graded.

Friday August 12: Last class day. Section 8.1 (page 227), #1, 2, 3*, 5.
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 not assigned, but are worth a look.

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