Math 12b -- Fall 2006
Linear Algebra and Discrete Dynamical Systems


Professors:
Nicholas J. Pippenger, Olin B165, x71114, my three initials  @ math.hmc.edu
Francis Su, Olin 1269, x73616, my lastname  @ math.hmc.edu

Office Hours: Prof. Su is available MONDAYS 4-6pm, Prof. Pippenger is available THURSDAYS 3-5pm.
Each of these times is open to ALL students from any section.

AE Hours: Tuesday, Thursday, Sunday evenings 8-10pm at the LAC, Riggs Room

Course content: This course is an introduction to linear algebra and discrete dynamical systems. Linear algebra is the study of linear functions of several variables. Matrices and matrix algebra are used to study and model them. Many real-world phenomena are modeled by functions of several variables, and it is often interesting and important to understand the long-term behavior of the "dynamics" of repeatedly iterating such functions. Discrete dynamical systems is the study of the dynamics of both linear and non-linear functions. We will study the dynamics of linear maps of several variables, as well as the non-linear maps of just one variable, whose dynamics are already quite interesting. You will see connections to calculus (Math 11) as well as see common themes with your later study of differential equations (Math 13) and multivariable calculus (Math 14).

Text: Linear Algebra: A Modern Introduction, 2nd edition by David Poole (ISBN 0-534-34174-8). Doing the reading will be essential for success in this course. We will sometimes be assigning homework from sections we have not yet covered so that you will read ahead.

Homeworks, Exams, and Grading:

Honor Code: Cooperation on homework assignments is encouraged, but each student is expected to write up his or her own solutions individually. (That is, no copying.) Comprehension is the goal, so you should understand solutions well enough to write them up yourself. In addition, you must cite any sources of help that you use. If you work with a classmate on a problem, be sure to acknowledge that person in your homework write-up. Harvey Mudd's honor code applies in all matters of conduct concerning this course.

Homework
(Due date)    
Assignment (* means optional reading or extra credit problems)
HW #1
(Fri 10/27)
Introduction to Vectors, The Dot Product
  • Read Sections 1.0-1.3 (1.4*) and Sections 3.0-3.1
  • Do Section 1.1 ( 4, 10, 12, 14, 16, 18 [Follow Good Problem instructions: Laying Out the Problem])
    1.2 ( 2, 8, 14, 20, 24, 44 ) 1.3 ( 2, 9 ) 1.4 ( 52* )

    You need only follow the "Good Problem" instructions for the specific problem marked "Good Problem", but you should think about good writing for all your homework problems!

  • HW #2
    (Tue 10/31)
    Matrix Operations, Matrix Algebra
  • Read Sections 3.2, 3.3 (to p.168), 2.0 - 2.2
  • Do Section 1.2 ( 54 [Follow Good Problem instructions: Flow], 62) 1.3 ( 14 ) 3.1 ( 2, 8, 10, 16, 18, 19, 20, 22, 32*, 38 )
  • HW #3
    (Fri 11/3)
    Systems of Linear Equations
  • Read Sections 2.3, 2.4
  • Do 3.2 ( 4, 29 [Follow Good Problem: Logical Connectives], 30b, 32*) 3.3 ( 4, 12, 17a, 43 ) 2.1 ( 16, 31, 42 ) 2.2 ( 12, 16, 36 )
  • HW #4
    (Tue 11/7)
    Inverses and Invertible Matrix Theorem
  • Read Section 3.3
  • Do Section 2.2 ( 19, 45, 60* )
    2.3 ( 2, 8, 10, 15, 18, 26, 36, 43, 46 [Follow Good Problem instructions: Introductions and Conclusions]),
    2.4 ( 5, 9 )
    In the Good Problem, your intro should explain your strategy of your proof, enabling the reader to follow your arguments easily, and your conclusion should summarize what you did.
  • HW #5
    (Fri 11/10)
    Subspaces, Linear Transformations
  • Read Sections 3.5, 3.6
  • Do Section 3.3 ( 23 [Follow Good Problem instructions: Mathematical Symbols], 26, 36, 38, 42, 46, 57, 59 )
    3.5 ( 4, 6, 10, 11, 12 )
    In the Good Problem, be sure to show your work and justify all steps.
  • HW #6
    (Tue 11/14)
    Determinants
  • Read Sections 4.1, 4.2
  • Do 3.5 ( 14, 16, 17, 21, 25, 32, 35, 39, 41, 48 )
    3.6 ( 12, 21, 31, 37, 50 )
  • Midterm
    Feedback!
    We invite your feedback on our teaching and your learning. Your responses remain anonymous. Constructive and specific suggestions to improve your learning experience are more valuable than vague comments.
    Thanks! Please use these forms:
  • Feedback for Professor Pippenger
  • Feedback for Professor Su
  • HW #7
    (Fri 11/17)
    Eigenvalues, Eigenvectors
  • Read Section 4.3 and review for your Take-Home Exam, to be handed out Friday, in class
  • Do 4.1 ( 4, 10, 17, 23, 25, 36 )
    4.2 ( 6, 13, 15, 23, 26, 27 )
    4.3 ( 18, 19a, 21 )

    p.s. it's a good idea when you're done to check if your answers seem reasonable. Answers to odd-numbered exercises can be found in the back of the book.

    The AE Tutors will be holding a REVIEW SESSION on THURSDAY evening... watch your e-mail for details.

  • TAKE-HOME EXAM
    (due MON 11/20)
    Take-home exams will be handed out Friday in class, due back Monday in class.
    HW #8
    (Tue 11/28)
    Discrete Dynamical Systems
  • Read your notes from class
  • Do Problems 1, 2, 3 from the Discrete Dynamical Systems Homework. In problem #2, when it asks you to give numerical estimates for any "limiting values" you detect, it just means to observe if the system tends to approach a fixed or periodic orbit, and if so, estimate the points in the orbit.

    This applet http://learn.sdstate.edu/cogswelk/homepage/WebDiagram/WebDiagram.html can be used to do the homework problems. While it doesn't have as many bells and whistles as Discrete Tool (below), it does run on Mac's as well as PC's, and does do graphical analysis.

    Discrete Tool may be found under the Start Menu in the PC labs, under the folder for ODE Architect.

  • HW #9
    (due Fri 12/1)
    Discrete Dynamical Systems
  • Read your notes from class
  • Do Problems 4, 5, 6, 7 from the Discrete Dynamical Systems Homework.
  • HW #9
    (due Tue 12/5)
    Discrete Dynamical Systems
  • Read your notes from class
  • Do Problems 9a, 11, 13, 15, 18 from the Discrete Dynamical Systems Homework.

    There are many websites that discuss bifurcation diagrams and the logistic map, e.g., Mathworld or various course websites.

  • HW #9
    (due Fri 12/8)
    Discrete Dynamical Systems
  • Read your notes from class
  • Do Problems 20, 22, 23, 25 from the Discrete Dynamical Systems Homework.
  • We'll be posting HW's a week in advance, so be sure to check the due date to ensure you are doing the correct assignment!