Mathematics 309, Sec. 502, Fall 2018

Linear Algebra for Differential Equations

Last updated Mon 17 Dec 2018

SKIP PAST THE ANNOUNCEMENTS if you so desire ._. (or to near the bottom)

ANNOUNCEMENTS

• 17 Dec: You should now be able to see your grade in eCampus. Letter grades are treated by eCampus in a weird way (apparently on a scale from 1 to 5 instead of 0 to 4), so the statistics it reported are more confusing than helpful. The grade distribution is 5 As, 10 Bs, 3 Cs, 4 Ds, 3 Fs, and 3 Qs. To get there required a massive curve: 16 points added to the total of two midterm tests and 44 points to the final exam, thereby raising all the final grades by one letter. That produces the sort of distribution I expect for a course at this level. I wasn't surprised that 3.5 weeks was not enough to instill mastery of Fourier series, and I was happy that most of you mastered orthogonal polynomials on the second try, but I was astonished that so many still don't understand what the dimension of a space of matrices is. Please study the solution key below, especially that first problem! The homework scores were scaled linearly to a max of 170.
• 8 Dec: Homework 13 grades are posted. I expect to be in my office most of Monday, in case you want to get your paper. I gave problems 1, 2, and 5 ten points each and problems 3 and 4 ten points total, since they were so similar and closely related.

  The total homework scores will be scaled to fit the points allowed in the grading scheme (details to be announced).

  When (as in problems 3, 4, 5 of this assignment) you need to leave a Fourier coefficient as an integral, it is not necessary to plug the integral formula into the series formula for the final solution. In fact, the solution is more readable (IMHO) if you write the coefficient formulas as separate equations. If you DO plug it in, change the variable of integration to a different letter (or attach a prime or tilde); otherwise there is an ambiguity. For example, in

  [sum] b_n sin(nx) [integral] sin(nx) f(x) dx

  the x inside the integral is not the same as the x outside; simplifying to [sin(nx)]^2 would be wrong!

• 6 Dec: Interim report on Homework 13: Before the final exam, please revisit problem 2, which involves a Fourier cosine series, and take note of the following:
  • In this case there is a term with n = 0. ( cos(0x) = 1 satisfies the homogeneous boundary conditions of this problem.)
  • You can't find the coefficient of that term from the formula you found for general n, because that derivation involved division by n, which is 0 here. Evaluate the (very easy) relevant integral separately. The general conclusion that a_n = 0 for even n does not apply to this term.
  • Oh, you didn't get a_n = 0 for even n? Check your integration and remember to include the contribution from the lower limit. ( cos(0) = 1, not 0.)
  • In problem 1 (and others), where both even and odd n do contribute, remember that the final answer is a single series. Don't write two separate equations for u(x,t) for even and odd n as if they were separate instances of the problem; add the two series together.
  I will have the homework graded by Monday.
• 6 Nov: The office hour on Thursday, Nov 15 is cancelled.
• 18 Oct: Next Monday's office hour (Oct 21) is cancelled.
• 9 Oct: Tomorrow's office hour will start at 3, not 2 (Wed 10 Oct).
• 28 Sep: The linear algebra help session is now Tuesday and Wednesday only (BLOC 161, 6-8 p.m.).
• 7 Sep: Next Math Club meeting, Sept. 11, 7:00 p.m.
• 4 Sep: Next AMUSE meeting, Sept. 5, 6:00 PM
• 31 Aug: The math department has announced a linear algebra help session in Blocker 161 from 6 to 8 p.m. every Tuesday, Wednesday, and Thursday. (Thursday was later cancelled.)
• 30 Aug: There is now a discussion forum in eCampus if you want to chat.
• 21 Aug: Yes, we have class on Thursday, Aug. 30. Official advice
• 15 Aug: Please read the course handout (linked below) when it becomes available. Meanwhile, here is some basic information:
  The main textbook is
  • S. J. Leon, Linear Algebra with Applications, 9th edition.
  For the last third of the course (covering Fourier series and partial differential equations) some other (inexpensive) resources will be provided or suggested.
  The class meets TR 3:55-5:10 in BLOC 163.

COURSE PROCEDURES AND RESOURCES

• Course handout
• Instructor: S. A. Fulling
  Office: Blocker 620H
  Email: fulling@math.tamu.edu
  Please see my home page for up-to-date office hours.
• Track your grades on eCampus.
• Groups
• Lecture notes:
  1. My old Math. 311 notes, published as a textbook (Linearity)
  2. Professor Vogel's notes on Fourier series and Bessel functions
  I will frequently, but not always, follow these in class. (This page of lecture notes contains material copyrighted by third parties, so you need to log in using your TAMU NetID username and password. If you have a problem, probably I mistyped your username in the password file; e-mail me.)

WEEKLY SCHEDULE AND ASSIGNMENTS

1. Aug 27-31: (1) The vector concept (2) Systems of linear equations

   Readings

   • Linearity Secs. 1.1 and 1.3. (There is nothing in Leon that precisely corresponds to these.)
   • Leon Sec. 1.1, and start on Sec. 1.2.

   Homework 1 (due Thursday, Sept. 6)

   1. From Leon Sec. 1.1 (p. 10): 1(b), 4(b), 5(b).
   2. From Leon Sec. 1.1 (p. 10): 6(a,f,h). (Use row reduction of augmented matrices, of course.)
   3. Essay question: A paleontologist said in a lecture, "In 10,000 BCE, at Lake Baikal the temperature rose 5 degrees Centigrade within 100 years." To make this more meaningful to the general public, a newspaper reporter translated it as, "Twelve thousand years ago, at Lake Baikal the temperature rose 41 degrees Fahrenheit within 100 years." What mistake did the reporter make, and what is the correct number?

2. Sep 3-7: Matrices

   Readings: Leon Secs. 1.2, 1.3, 1.4, and pages 64-66 in Sec. 1.5

   Homework 2 (due Thursday, Sept. 13)

   1. Leon Sec. 1.2 (pp. 23-26): 5(c,d,i,k), 10, 21
   2. Leon Sec. 1.3 (pp. 45-46): 1, 9(a,b)
   3. Leon Sec. 1.4 (pp. 58-60): 1, 7, 30
   4. Leon Sec. 1.5 (p. 69): 10(a,b,d,e,f) (You don't need to learn "elementary matrices" from Sec. 1.5 to do these. Use the row-reduction method taught in class.)

3. Sep 10-14: (1) Determinants (2) Definition of a vector space

   Readings

   • Leon Chap. 2. (Linearity Sec. 2.5 will tell you less about determinants than Leon; Secs. 7.1 and 7.2 will tell you more.)
   • Leon Sec. 3.1.

   Homework 3 (due Thursday, Sept. 20)

   1. Leon Sec. 2.1 (p. 94): 4(a,b,c)
   2. Leon Sec. 2.2 (p. 100): 2(a), 12
   3. Leon Sec. 2.3 (p. 109): 15(d), 16(a)
   4. Leon Sec. 3.1 (p. 122): 5, 6, 11

4. Sep 17-21: Spanning and linear independence

   Readings: Leon Secs. 3.2 (emphasis on spanning) and 3.3

   Homework 4 (due Thursday, Sept. 27)

   1. Leon Sec. 3.2 (p. 129): 11(a,b,d), 12(a,b,e), 14, 15
   2. Leon Sec. 3.3 (pp. 140-141): 1(a,b,d), 2(a.b.e), 4, 5, 8(a,c), 11

5. Sep 24-28: Bases

   Readings: Leon Secs. 3.4 and 3.5

   Homework 5 (due Thursday, Oct. 4)

   1. Leon Sec. 3.4 (p. 146): 3, 5(a,b,c), 8, 10
   2. Leon Sec. 3.5 (pp. 156-157): 1(a,c), 2(b,c), 3(a), 5(a,b(i)), 10

6. Oct 1-5: (1) Linear transformations (2) TEST (not including Secs. 3.5, 3.6, and 4.1)

   Readings: Leon Sec. 4.1

   Homework 6 (due Thursday, Oct. 11)

   1. Leon Sec. 4.1 (p. 177): 1(a,d,e), 3, 9(a,b), 10

7. Oct 8-12: Matrices of linear transformations and changes of basis

   Readings: Leon Secs. 4.2 and 4.3

   Homework 7 (due Thursday, Oct. 18)

   1. Leon Sec. 4.2 (pp. 190-192): 2(a), 3, 5(a,b), 6, 13, 14(a), 15, 18(c)
   2. Leon Sec. 4.3 (pp. 197-198): 1(a,d), 2, 6, 15

8. Oct 15-19: Subspaces

   Readings:

   • Leon Secs. 3.2 (review), 3.6
   • Linearity Sec. 5.3

   Homework 8 (due Thursday, Oct. 25)

   1. Leon Sec. 3.2 (pp. 128-129): 1. 3(a,d,e,f,g), 6(a,e), 8, 22
   2. Leon Sec. 3.6 (pp. 162-164): 1(a,b), 2, 4(a,b,d,e)), 16, 21, 25
   3. Leon Sec. 4.1 (p. 178): 17, 19, 23, 25

9. Oct 22-26: Inner products and orthogonality

   Readings: Leon Chapter 5. (Omit the 3 applications on pp. 209-215, and Sec. 5.3, and pp. 252-263 except for the 5 properties at top of p. 252. We will come back to pp. 258-263 later.) Leon tells us more about inner products than we have time to absorb in this course; use Linearity Secs. 6.1-6.2 and the lectures based on them as a guide to what I consider most important.

   Homework 9 (due Thursday, Nov. 1)

   1. Leon Sec. 5.1 (p. 216): 13
   2. Leon Sec. 5.2 (p. 224): 2
   3. Leon Sec. 5.4 (pp. 246-247): 3, 5, 6, 9, 12, 13, 25, 28
   4. Leon Sec. 5.5 (pp. 263-264): 4, 8
   5. Leon Sec. 5.6 (p. 274): 3, 4
   6. Leon Sec. 5.7: 2 (use Gram-Schmidt, not Thm. 7.2)

10. Oct 29 - Nov 2: Eigenvectors and diagonalization

    Readings: Leon Chapter 6, except Secs. 6.5, 6.7, 6.8

    Homework 10 (due Thursday, Nov. 8)

    1. Leon Sec. 6.1 (p. 299): 1(a,c,d,g,h,j), 7, 8
    2. Leon Sec. 6.2 (p. 310): 2(a) (Use the matrix method, of course.)
    3. Leon Sec. 6.3 (p. 327): 1(a,d), 2(a,d), 3(a,c,d)
    4. Leon Sec. 6.4 (p. 340): 5(a,f)
    5. Leon Sec. 6.6 (p. 369): 3(a,c,d), 7(a,b,e)

11. Nov 5-9: (1) Introduction to PDEs (2) TEST (through Chapter 6)

    Readings: Vogel Chapters 1 and 2

    Homework 11 (due Thursday, Nov. 15)

    1. Vogel Ch. 1 (p. 6): 1, 2
    2. Vogel Ch. 2 (p. 9): 1(c,d), 2, 3

12. Nov 12-16: Fourier series basics

    Readings:

    • Vogel Chapter 3
    • Leon pp. 258-263

    Homework 12 (due Tuesday, Nov. 27): Vogel Ch. 3 (p. 15), all

13. Nov 19-28: Fourier series solutions

    Readings: Vogel Chapter 4

    Homework 13 (due Tuesday, Dec. 4): Vogel Ch. 4 (p. 19), all

14. Nov 29 - Dec 5: Bessel functions

    Readings: Vogel Chapter 5

    Homework: You're welcome to try pp. 25-26, but nobody will grade it.

TEST SOLUTIONS

• Test A
• Test B
• Final Exam

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