Math320

Spring 2012:
Math 320 Section 001: Linear Algebra and Differential Equations

 

Schedule: MWF 9:55 – 10:45 AM. Van Vleck B 102

Syllabus: Can be found here

Textbook: Edwards and Penney, Differential Equations and Linear Algebra, third ed., Prentice Hall

Teaching assistants webpages for this course:

Sections 301, 302: Hesamaddin Dashti

Sections 303-306: Erkao Bao

 

Homework assignments:

Hw 1 Due Wednesday, February 1st. Solutions

Hw 2 Due Wednesday, February 8th. Solutions

Hw 3 Due Friday, February 17. Solutions

Hw4 Due Wednesday February 22. Solutions

Hw5 Due Friday March 2. Solutions

Hw6 Due Friday March 9. Solutions

Hw7 Due Friday March 16. Solutions

Hw8 Due Wednesday, March 21. Solutions

Hw9 Due Monday, March 26 Solutions

Hw10 Due Friday, April 13 Solutions

Hw11 Due Friday, April 20 Solutions

Hw12 Due Friday, April 27 Solutions

Hw13 Due Friday, May 4 Solutions

Hw 14 Optional

 

Practice Exam You can now find the solution HERE

 

Practice Exam 2 You can now find the solution HERE

 

Practice Exam 3 You can now find the solution HERE

 

 

Brief lecture outline:

 

Lecture 1: January 23

  1. Section 1.1 Differential equations and mathematical models

  2. Newton's law of cooling

  3. Population growth

 

Lecture 2: January 25

  1. Section 1.2: Integrals as general and particular solutions

  2. Second order Equations

  3. Velocity and acceleration

 

Lecture 3: January 27

  1. Section 1.3 Slope fields and solution curves

  2. Graphical method

  3. Existence and uniqueness of solutions

 

Lecture 4: January 30

  1. Section 1.4 Separable equations and applications

  2. Implicit, general and singular solutions

  3. Cooling and heating

 

Lecture 5: February 1

  1. Section 1.5 Linear first order equations

  2. Integrating factors

  3. Mixture problems

 

Lecture 6: February 3

  1. Section 1.6 Substitution methods and exact solutions

  2. Homogeneous equations

 

Lecture 7: February 6

  1. Bernoulli equations

  2. Exact differential equations

  3. Reducible second-order equations

 

Lecture 8: February 8

  1. Chapter 2: Mathematical Models and Numerical Analysis

  2. Section 2.1 Population models

  3. Bounded populations and the logistic equation

  4. Section 2.2 Equilibrium solutions and stability, autonomous equations

 

Lecture 9: February 10

  1. Logistic population with harvesting

  2. Bifurcation and dependance on parameters

  3. Section 2.4 Numerical approximation: Euler's method

 

Lecture 10: February 13

  1. Local and cumulative errors

  2. Section 2.5: A closer look at Euler's method

  3. Improved Euler's method

 

Lecture 11: February 15

  1. Chapter 3: Linear systems and matrices

  2. Section 3.1: Introduction to linear systems

  3. Two equations in two unknowns

  4. The method of elimination

  5. Three equations in three unknowns

 

Lecture 12: February 17

  1. Section 3.2: Matrices and Gaussian elimination

  2. Elementary row operations. Row equivalent matrices

  3. Gaussian elimination method

 

Lecture 13: February 20

  1. Echelon matrices

  2. Leading and free variables

  3. Section 3.3: Reduced row-echelon matrices

 

Lecture 14: February 22

  1. Gauss-Jordan elimination

 

Lecture 15: February 24: Exam 1

 

Lecture 16: February 27

  1. Homogeneous linear systems

  2. Section 3.4: Matrix operations

  3. Multiplication of matrices

 

Lecture 17: February 29

  1. Multiplication of matrices. Matrix algebra

  2. Section 3.5: Inverse of matrices.

 

Lecture 18: March 2

  1. Algorithm for finding A^{-1}

  2. Section 3.6 Determinants

  3. 2x2 and nxn determinants

 

Lecture 19: March 5

  1. Row and column properties for determinants

  2. Cramer's rule for nxn systems

 

Lecture 20: March 7

  1. Chapter 4: Vector spaces

  2. Addition and multiplication by scalars

  3. Collinear vectors. Linear dependance

  4. Linear independence in R^3

 

Lecture 21: March 9

  1. Basis vectors in R^3

  2. Subspaces of R^3

 

Lecture 22: March 12

  1. Section 4.2: The vector space R^n and subspaces

  2. Addition and multiplication by scalars. Properties of vector spaces.

  3. The vector space of real valued functions

 

Lecture 23: March 14

  1. Subspaces

  2. Section 4.3: Linear combinations and independence of vectors

  3. Linear span

  4. Linear independence.

 

Lecture 24: March 16

  1. Section 4.4 Bases and dimension for vector spaces

  2. Basis

  3. Finite dimensional spaces

 

Lecture 25: March 19:

  1. Bases for solution spaces

  2. Algorithm for find a basis by elementary row operations

 

Lecture 26: March 21

  1. Section 4.5: Row and column space

  2. Row and column rank

  3. Extracting or completing bases

 

Lecture 27: March 23

  1. Chapter 5: Higher order linear differential equations

  2. Section 5.1 Intro: 2nd order linear equations

  3. Homogeneous equations

  4. A typical application

  5. Linear combinations: General solutions

 

Lecture 28: March 26

  1. Wronskians and linear independence

  2. Linear 2nd order equations with constant coefficients

  3. Case: Distinct roots

  4. Case: Repeated roots

  5. Section 5.2: General solutions of linear equations

 

Lecture 29: March 28

  1. Midterm 2

 

Lecture 30: March 30

  1. Non-homogeneous equations

  2. Particular and complementary solutions

  3. Section 5.3: Homogeneous equations with constant coefficients

  4. The characteristic equation

 

Lecture 31: April 09

1. Case: Distinct roots

2. Case: Repeated roots

 

Lecture 32: April 11

1. Case: Complex roots

2. Case: Repeated complex roots

 

Lecture 33: April 13

1. Section 5.4: Mechanical vibrations

2. Application: The simple pendulum

3. Free undamped motion

 

Lecture 34-36: April 16,18,20 (Covered by Uri Andrews) Sections 5.5, Section 5.6 (pages 353-357) and started 7.1

 

Lecture 37: April 23

  1. Chapter 7: Linear systems of differential equations

  2. Section 7.1: First order systems and applications

  3. Simple two dimensional systems

 

Lecture 38: April 25

  1. Section 7.2: Matrices and linear systems

  2. Homogeneous equations

 

Lecture 39: April 27

  1. Homogeneous equations

  2. Wronskians and linear independence

  3. General solutions

  4. Non-homogeneous solutions and the superposition principle

 

Lecture 40: April 30

  1. Section 7.3: The eigenvalue method for linear systems

  2. Eigenvalues and eigenvectors

  3. The characteristic polynomial

 

Lecture 41: May 2

  1. Eigenvalues, eigenvectors and linearly independent solutions

  2. The eigenvalue method

  3. Distinct real eigenvalues

  4. Complex eigenvalues (compartmental analysis has been excluded, as well as applications)

 

Lecture 42: May 4

  1. Section 7.5 Multiple eigenvalue solutions (section 7.4 has been excluded)

  2. Repeated eigenvalues, multiplicity

  3. Complete eigenvalues (defective eigenvalues have been excluded)

 

Lecture 43: May 7

  1. Chapter 8: Matrix exponential methods

  2. Section 8.1: Matrix exponentials and linear systems

  3. Fundamental matrices

  4. Solving initial value problems with the fundamental matrix

 

Lecture 44: May 9

  1. Matrix exponential solutions

  2. Initial value problems and the exponential solution

  3. How to compute the exponential of a matrix: for nilpotent matrices, using the fundamental matrix and by diagonalizing the matrix.

 

Lecture 45: May 11

  1. Examples: Computation of exponential matrices

  2. Review session

 

 

Discussion sessions:

 

Section

Time

Room Number

Teaching Assistant

Email

301

T 8:50-9:40AM

B119 Van Vleck Hall

Hesamaddin Dashti

dashti at math dot wisc dot edu

302

R 8:50-9:40AM

B119 Van Vleck Hall

Hesamaddin Dashti

dashti at math dot wisc dot edu

303

T 11:00-11:50AM

115 Ingraham Hall

Erkao Bao

bao at math dot wisc dot edu

304

R 11:00-11:50AM

115 Ingraham Hall

Erkao Bao

bao at math dot wisc dot edu

305

T 9:55-10:45AM

B329 Van Vleck Hall

Erkao Bao

bao at math dot wisc dot edu

306

R 9:55-10:45AM

B329 Van Vleck Hall

Erkao Bao

bao at math dot wisc dot edu

 

 

Other important information:

 

Exam 1: February 24 Time: 9:55 - 10:45 AM. Room: Van Vleck B 102 20% of the Final Grade

Exam 2: March 28 Time: 9:55 - 10:45 AM. Room: Van Vleck B 102 25% of the Final Grade

Exam 3: May 13 Time: 7:45 – 9:45 AM. Room: Van Vleck B 102 30% of the Final Grade

 

*See the syllabus for more information on exam policies.

 

The TAs will grade a subset of the homework problems given out each week (with some points also given for completeness). The homework scores will count for 15% of the grade. The lowest homework score will be dropped.

There will be an estimate of 6 quizzes, to be scheduled during section meetings on dates to be determined by the TA. Quizzes will be graded and will count for 5% of the overall grade. The lowest quiz score will be dropped. There will be no make-up quizzes.

Class participation: 5 % of the final grade

 

Grading:

Exam 1: 20 %

Exam 2: 25 %

Exam 3: 30 %

Homework: 15%

Quizzes: 5%

 

Class participation: 5%

 

Note: Any student with a documented disability should contact me as soon as possible so that we can discuss arrangements to fit your needs.