## Rose-Hulman - Department of Mathematics - Course Syllabus MA212 - Matrix Algebra and Systems of Differential Equations - 2010-11

 Description & Prerequisites Course Goals Texts and Other Materials Course Topics Course Policies Links last posted to site: 05/29/12

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 (Rose-Only) Mathematics Course Information Repository.

### Course Goals

1. Develop a deeper understanding of systems of equations and their solutions, especially linear algebraic and differential equations.
2. Provide an introduction to Fourier series.
3. Develop a deeper understanding and appreciation of transformation and approximation methods by studying Fourier methods.
4. Improve mathematical modeling and analytical problem solving skills.
5. Develop ability to communicate mathematically.
6. Improve skill using the computer as a tool for mathematical analysis and problem solving.
7. 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.

### Course Topics

The topics below follow the approximate book order.

1. 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.
2. First order linear differential equations
• Separable equations
• Linear equations

3. 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 points
• 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

4. 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:

#### Computer Policy

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.)

#### Performance Standards

##### Paper and pencil
1. Demonstrate skill with matrix arithmetic including
• 2x2 inverses
• eigenvalues and eigenvectors for 2x2 matrices
2. Demonstrate skill using complex numbers
• basic arithmetic and geometric interpretation
3. Solve linear differential equations including
• direct integration
• separate and integrate involving simple integrations
• integrating factors with simple integration
• graphing and interpreting the solutions
4. Linearize systems of non-linear equations, determining and classifying critical points
5. Sketch  phase portraits of simple systems
6. Compute Fourier coefficients of simple functions
##### Maple
1. matrix algebra
• solve linear systems of equations
• compute eigenvalues and eigenvectors
2. 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.