Math 2243: Linear Algebra and Differential Equations 

Lecture 020, Fall 2004

Science Classroom Building 175, 11:15 am-12:05 pm MWF

Contact Information for the Instructor:  

Instructor: Willard Miller
Office: Vincent Hall 513
Office Hours: 10:10-11:00 MW, 12:20-1:10 F,  or  by appointment  
 Phone: 612-624-7379
miller@ima.umn.edu, miller@math.umn.edu
www.ima.umn.edu/~miller/

 Discussion Sections:

                 -021  10:10am-11:00am TTh,   VinH 311       Matthew Dobson  Phone:  5-5099,   Office:  VinH 420,  dobson@math.umn.edu

                 -022  10:10am-11:00am TTh,   VinH 211       Ji Hoon Ryoo,  Phone:  5-8553, Office: VinH 422,   jhryoo@math.umn.edu

                 -023  11:15am-12:05pm TTh,   FolH 334       Matthew Dobson  Phone:  5-5099,   Office:  VinH 420,  dobson@math.umn.edu

                 -024  11:15am-12:05pm TTh,   VinH 311        Ji Hoon Ryoo  Phone:  5-8553, Office: VinH 422,   jhryoo@math.umn.edu


Brief Course Description
Credit will not be granted if credit has been received for: MATH 2373, prereq 1272 or 1282 or 1372 or 1572,  4 credits

Overview: The course is divided into two somewhat related parts.
Linear algebra: matrices and matrix operations, Gaussian elimination, matrix inverses, determinants, vector spaces and subspaces, dependence, Wronskian, dimension, eigenvalues, eigenvectors, diagonalization.
ODE: Separable and first-order linear equations with applications, 2nd order linear equations with constant coefficients, method of undetermined coefficients, simple harmonic motion, 2x2 and 3x3 systems of linear ODE's with constant coefficients, solution by eigenvalue/eigenvectors, nonhomogenous linear systems; phase plane analysis of 2x2 nonlinear systems near equilibria.

Audience: Part of the standard 2nd year calculus course for students outside of IT.

Text: Farlow, Hall, McDill, West. Differential Equations and Linear Algebra   We will cover Chapters 1-7 (up to Section 7.2) as well as Sections 8.1, 8.2 and 8.5.

4 credits. 3 lectures, 2 recitations per week.
Information about Course Management

Final Grade: Based on a possible 700 points
        Homework and quizzes in recitation sessions -- 100 points
        Three midterms (100 points each: Sept. 28, Oct. 26, Dec. 2) -- 300 points
        Final Exam (Dec. 16)-- 300 points
Effort required: About 12 hours a week
Required Homework: Assigned in recitation session; a subset of the suggested homework.

Calculator and Exam policy: A calculator is useful for homework. Calculators will be allowed but no books or notes in the midterms and final exams.
Makeup policy:  No makeup tests without rigorous emergency reasons. Athletes please present "Proofs of Activities" in advance.
Drop Dates: Students may drop the course without permission before the end of the 8th week of the semester (November 1). If you drop before the end of the 2nd week, no mention of the course appears on your transcript. Otherwise, you receive a "W." Starting November 2,  you need permission to drop the course.
Student Conduct:  Statement on Scholastic Conduct: Each student should read the college bulletin for the definitions and possible penalties for scholastic dishonesty.  Students suspected of cheating will be reported to the Scholastic Conduct Committee.


FINAL EXAM:  1:30 - 4:30 pm, Thursday, December 16

"All students must have their official University I.D. Card with them at the time of the final exam and must show it to one of the proctors when handing in their exam. The proctor will NOT accept a final exam from a student without an I.D. Card." 


   Syllabus and Suggested Homework

     Date                                Lecture                                    Suggested Homework Problems

Wednesday, September 8
 Section 1.1 Dynamical Systems, Modeling
 #23-27
Friday, September 10
 Section 1.2 Solutions and Direction Fields
 #2,3,6,7,13-18,20,23,24,26,29,30
Monday, September 13
 Section 1.3 Separation of Variables: Quantitative Analysis
 #12,13,16-26,29,31,32,39,40,43-47,53,54
Wednesday, September 15
 Section 1.4 Euler's Method: Numerical Analysis
 #1,7-10,12
Friday, September 17
 Section 1.5 Picard's Theorem: Theoretical Analysis
 #2-4,7-12,21,23
Monday, September 20
 Section 2.1 Linear Equations: The Nature of Their Solutions
 #7-12
Wednesday, September 22
 Section 2.2 Solving the 1st-Order Linear ODE
 #1-6,9-15,19,20,25,26,30-35,45,46
Friday, September 24
 Section 2.3 Growth and Decay Phenomena
 #1-5,7,14,16,17,21,24,26,33
Monday, September 27
 Review
Tuesday, September 28
 Midterm I
Wednesday, September 29
 Section 2.4 Linear Models: Mixing and Cooling
 #2,4,7,8,11-13,17-19
Friday, October 1
 Section 2.5 Nonlinear Models: Logistic Equation
 #13-18,21-24
Monday, October 4
 Section 2.6 Systems of Differential Equations: A First Look
 #1-3,5-7,11,14-17,23
Wednesday, October 6
 Section 3.1 Matrices: Sums and Products
 #4,5,31-34,40,42,44-46,51,54,61
Friday, October 8
 Section 3.2 Systems of Linear Equations
 #5-10,29-33,35
Monday, October 11
 Section 3.3 The Inverse of a matrix
 #7-10,14,17,18,24,25
Wednesday, October 13
 Section 3.4 Determinants and Cramer's Rule
 #1,4-7,11,12,16-18,20,23,29-32,34-36
Friday, October 15
 Section 3.5 Vector Spaces and Subspaces
 #2,4-6,8,9,12-14,16-20,22,23,25,26,34,36-38,43-51,56
Monday, October 18
 Section 3.6 Basis and Dimension
Wednesday, October 20
 Section 3.6
 #1,2,5-8,11-14,16-18,20-22,26-32,38,39,42-45,47-50,54-58,64,65
Friday, October 22
 Section 4.1 The Harmonic Oscillator
 #4-6,8-10,14-16,21,22,25-28,31-39,49,52
Monday, October 25
 Review
Tuesday, October 26
 Midterm II, Covers material through Section 3.6
Wednesday, October 27
 Section 4.2 Real Characteristic Roots
 #16-24,28-30,35,38-41,43,47,55,56,59-62
Friday, October 29
 Section 4.3 Complex Characteristic Roots
 #13,15-28,34-36,41,42
Monday, November 1
 Section 4.4 Undetermined Coefficients
 #4-6,22-33,36,39,40
Wednesday, November 3
 Section 4.5 Forced Oscillations
 #4-13,15-17,20,22,23
Friday, November 5
 Section 4.6 Conservation and Conversion
 #1,2,8-13,17-20,32,33,40,41,44
Monday, November 8
 Section 5.1 Linear Transformations
 #6-10,13-15,17-29,33,34,37-43,47,59,61-63,65-67,70-73,78-80
Wednesday, November 10
 Section 5.2 Properties of Linear Transformations
 #2,3,6,7,10-14,23,30,34,35,38,39,44-47,50-52,55-61
Friday, November 12
 Section 5.3 Eigenvalues and Eigenfunctions
 #6-9,15,18,20,22,24,26-29,31,33-36,39-41,45-47
Monday, November 15
 Section 5.4 Coordinates and Diagonalization
 #1-3,7-9,13-15,20-23,27-33,39-41,43-44
Wednesday, November 17
 Section 6.1 Theory of Linear DE Systems
 #3,5,7,9,17-19,23-26,33-35
Friday, November 19
 Section 6.2 Linear Systems with Real Eigenvalues
 #15-23,28,31,39,40,43
Monday, November 22
 Section 6.3 Linear Systems with Nonreal Eigenvalues
 #14-17,23-27,30,33
Wednesday, November 24
 Section 6.4 Decoupling a Linear DE System
 #3-5,9-11,15,16
Monday, November 29
 Section 6.5 Stability and Linear Classification
 #1-12,14,15,17
Wednesday, December 1
 Review
Thursday, December 2
 Midterm III, covers material through Section 6.4
Friday, December 3
 Section 7.1 Nonlinear Systems
 #11,13,15,17,19,21,23,27-33,35,37
Monday, December 6
 Section 7.2 Linearization
 #1-9,11-18,20-23
Wednesday, December 8
 Section 8.1 Linear Nonhomogeneous problems
 #3,5,11,13,15-17,24,25
Friday, December 10
 Section 8.2 Variation of Parameters
 #3-9,12-15,18,19
Monday, December 13
 Section 8.5  Chaos in Forced Nonlinear Systems
 didn't cover this
Wednesday, December 15
 Review
Thursday, December 16
 Final Exam 1:30-4:30 pm        Sec. 021: VinH 113,
  Secs. 022-023:   MurphyH 130,          Sec. 024:  MurphyH 214


PDF files of exams and practice exams from earlier versions of this course.
Note: These exams were written by different instructors and were sometimes based on different texts, so they should be used with care.

     Midterm1    Midterm 2    Midterm 3    Sample Final    Final

     Exam 1           Exam 2        Exam 3

    Midterm 1a    Midterm 2a    Midterm 3a    Midterm 4a   Final a

   Midterm 1b     Midterm 2b    Midterm 3b    Final b    Final b solns

   Final 1c solns

    My own practice exams and actual exams, based on the present course text.

    Practice Midterm1   Midterm1 solutions     Practice Midterm2    Midterm2  solutions       Practice Midterm3     Midterm3  solutions

     An example of a linear operator on a space of polynomials         Practice Final

Advisory grades for Midterm 1                

85-100  A                                                           Mean      59.46
80-84    A-                                                         Median   55
75-79    B+                                                         STDev    18.90
70-74    B                                                           Number 137
65-69    B-   
55-64    C+  
45-54    C  
40-44    C-  
35-39    D+ 
30-34    D   
< 30       N                                                                                                                                                                               


Advisory grades for Midterm 2  

95-100   A                                                        Mean       76.16
90-94     A-                                                      Median    80
85-89     B+                                                      STDev     20.7
80-84     B                                                        Number  134
75-79     B-
70-74     C+
65-69     C
60-64     C
55-59     C
50-54     C-
45-49     D+
40-44     D
35-39     D
<    34     N
  
Advisory grades for Midterm 3

80-100    A                                                    Mean       51
70-74      A-                                                  Median    48
65-69      B+                                                  STDev     17.8
55-64      B                                                    Number 127
50-54      B-
45-49      C+
35-44      C
30-34      C-
20-29      D
<   14       N


Miscellaneous Goodies

Plots of direction fields using MAPLE, showing some of the advantages and pitfalls of graphics software packages

            This page also shows a plot of the phase diagram for a linear system of differential equations, considered in class, and a 2D preditor-prey                     model, showing cyclical behavior around an equilibrium point. This particular preditor-prey model system doesn't have an explicit solution    in terms of standard functions. There is also a phase plane diagram for an underdamped harmonic oscillator showing the "black hole" at the     equilibrium point.

Spreadsheet with examples of the Euler method for approximating solutions of 1st order ODEs, including effect of step size

        There are two examples treated:
1. Solution of equation y'=t-3y from t=0 to t=1, with initial condition y(0)=1, step size h=.001     
2. Solution of equation y'=y/t -1  from t=1 to t=2, with initial condition y(1)=1, step sizes h=0.1, h=0.01, h=0.001. According            to theory,  decreasing step size by a  factor of 10  should decrease maximum discretization error by the same factor. However,  decreasing  step size by a factor of 10 may increase maximum roundoff error 10 times. In this example the total error, discretization  and roundoff, is listed in the righthand column. For h=0.1  the maximum error is about 5 x 10-2 , for h=0.01 the maximum error is about 5 x 10-5 , while for for h=0.001 the maximum total error is  5 x 10-4 . Thus reducing step size improves  accuracy initially, but eventually the increased roundoff error actually reduces the accuracy.                                    
        
Spreadsheet showing Future Value of $100 at 5% annual interest with various periods of compounding, including continuous compounding


Introduction to MATLAB (courtesy of Professor Peter Olver)      Postscript file           PDF file


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