Rose-Hulman - Department of Mathematics - Course Syllabus
MA212 - Matrix Algebra and Systems of Differential Equations - 2010-11
Parts of the web page to be completed or determined by the instructor are in green.
Catalogue Description and Prerequisites
MA 212 (starting Fall 2010) Matrix Algebra and Systems of Differential Equations 4R-0L-4C F,W,S Pre: MA 113
Basic matrix algebra with emphasis on understanding systems of linear equations from algebraic and geometric viewpoints, and eigenvalues and eigenvectors. Solution of systems of first order linear differential equations by eigensystems and investigation of their solution structure determined by eigensystems. Phase portrait analysis and classification of the nature of the stability of critical points for linear and nonlinear systems. Fourier series. Introduction to complex arithmetic, as needed. Applications to problems in science and engineering.
Prerequisite: All of the topics in MA112, as well some topics
in MA113, will be assumed. In addition, familiarity with some Maple commands
from Calculus is assumed. The Maple files MapleIntRGL.mws (introduction)
and calcrev.mws (review for DE) may be helpful. These files are available on
Angel in the
Course Information Repository.
- Develop a deeper understanding of systems of equations and their solutions, especially linear algebraic and differential equations.
- Provide an introduction to Fourier series.
- Develop a deeper understanding and appreciation of transformation and
approximation methods by studying Fourier methods.
- Improve mathematical modeling and analytical problem solving skills.
- Develop ability to communicate mathematically.
- Improve skill using the computer as a tool for mathematical analysis and
- Introduce applications of mathematics, especially to science and engineering.
Textbook and other required materials
Textbook: Text: Advanced Engineering Mathematics, 4th edition by Zill and Wright
Computer Usage: Maple14 must be available on your laptop.
The topics below follow the approximate book order.
- Matrix algebra
- Matrix Arithmetic (e.g., addition, scalar multiplication, matrix multiplication, inverses --emphasis on the 2 x 2 case for "by-hand" inverting)
- Understanding a matrix A as a transformation (avoiding general vector space ideas) ; Ax = b
- Representation of systems of linear equations as matrix equations
- Gaussian Elimination for solving Ax = b.
- Structure of general solution for Ax = b: algebraic point of view (particular solution plus homogeneous solution);
- Eigenvalues and Eigenvectors: algebraic point of view (solve Ax = lx for x and l ),
and geometric point of view (Ax is parallel to x)
- Applications as appropriate.
- First order linear differential equations
- Separable equations
- Linear equations
- Systems of first order differential equations
- Solution and solution structure determined by eigensystem, for x' = Ax and x' = Ax
+ b, (b constant and A invertible)
- Trajectories and phase portrait for x' = Ax and x' = Ax + b near critical
- Classification and stability of critical points for linear systems
- Phase portrait, linearization, and stability of critical points for non-linear systems
- Numerical solutions of systems
- Applications as appropriate - e.g., predator/prey and competing species for phase portraits,
tanks for eigenvalue methods
- Orthogonal Functions and Fourier Series
- Orthogonal functions
- Fourier Series
- Sine and cosine series
Course Requirements and Policies
The following policies and requirements will apply to all sections and classes:
A summary of the computer policy page: Students will
be expected to demonstrate a minimal level of competency with a relevant computer algebra system. The
computer algebra system will be an integral part of the course and will be used regularly in class work,
in homework assignments and during quizzes/exams. Students will also be expected to demonstrate the ability
to perform certain elementary computations by hand. (See Performance Standards below.)
Paper and pencil
- Demonstrate skill with matrix arithmetic including
Demonstrate skill using complex numbers
- 2x2 inverses
- eigenvalues and eigenvectors for 2x2 matrices
Solve linear differential equations including
- basic arithmetic and geometric interpretation
Linearize systems of non-linear equations, determining
and classifying critical points
Sketch phase portraits of simple systems
Compute Fourier coefficients of simple functions
- direct integration
- separate and integrate involving simple integrations
- integrating factors with simple integration
- graphing and interpreting the solutions
- matrix algebra
- solve linear systems of equations
- compute eigenvalues and eigenvectors
- solve and plot differential equations
- using dsolve
- numerically by using the method=numeric
- using Maple to assist in separate and integrate
- graphing and interpreting the solution
Final Exam Policy
The following is an extract from the final exam policy page. Consult the policy page for complete details. The final exam will consist of two parts. The first part will be "by hands" (paper, pencil). No computing devices (calculators/computers) will be allowed during the first part of the final exam. This part of the exam will cover both computational fundamentals as well as some conceptual interpretation, though the level of difficulty and depth of conceptual interpretation must take into account that this part of the exam will be shorter than the second part of the exam. The laptop, starting with a blank Maple work sheet, and a calculator, may be used during the second part of the exam. No "cheat sheets", prepared Maple worksheets or prepared program on the calculator may be used. The second part of the exams will cover all skills: concepts, calculation, modeling, problem solving, and interpretation.
Individual Instructor Policies
Your instructor will determine the following for your class:
*Note that most instructors will enforce some type of grade penalty for students with more than four unexcused absences.
- the grading scheme, based on the various course components.
- the number and format of hour exams, quizzes, homework assignments, in class assignments, and projects,
- the policies governing the work items above, e.g., whether the computer will be used, what collaboration is allowed, and the format of assignments.
- all policies for classroom procedure, including group work, class participation, laptop use and attendance*.