# Linear Algebra and Differential Equations

Linear Algebra and Differential Equations

Linear Algebra and Differential Equations are the fundamental mathematical tools for sophisticated
mathematical modeling and applications in the natural and social sciences. By now you have had significant
experience using algebra to formulate and solve problems -- particularly when there is only a single
variable to work with. You have also had some experience setting up and solving systems of equations
which involve more than one unknown. Through our study of linear algebra we will learn techniques for
computing with matrices and vectors. This turns out to be pretty handy in the study of differential equations
(and also in studying Multivariable Calculus). Differential equations are simply equations which involve
derivatives. You have been working with these in Calculus while developing your concept of derivatives
and techniques of integration. In this course, we will classify and learn to solve ordinary differential
equations: equations which involve derivatives of functions of just one dependent variable. We will see
some applications in the sciences, but we will focus on the mathematical theory, concepts, and tools that
are required to analyze and solve simple applications.

According to the course description in the university catalogue, our study will address the following topics
(although not necessarily in the order listed): linear systems, abstract vector spaces, matrices through eigenvalues
and eigenvectors, solutions of ordinary differential equations, Laplace transforms, and first order
systems.

Prerequisite

Successful completion of Differential and Integral Calculus is required for this course. This means that
you have taken and passed MATH 1220 or earned a score of 5 on a BC Calculus exam.

Materials

We will use the textbook called “Differential Equations: A Systems Approach” by Goldberg and Potter.
This means that group projects and homework exercises will be assigned from this book, and that I will
allow the authors’ approach to the subject guide ours. You may be interested in using a book called “Introduction
to Linear Algebra and Differential Equations” by Dettman (published by Dover) because it covers
what we will be studying in class and it only costs about \$15. This book is not required! We will also
be making use of MATLAB. You need to have a computer account and access to the labs which have
MATLAB installed (ENGR 305, Ag Sci 119). You may purchase a student version of MATLAB at the bookstore
for your own computer, but this is not required -- the cost is about \$100. You are free to use whatever
calculator you wish to use, but no calculators are allowed on in-class quizzes.

Assignments

We will assign and review homework exercises each class day, but I will not collect or grade these. A quiz
on homework problems will be given at the beginning of class each Friday. There will be 11 quizzes offered
in total, but your lowest score will be dropped to compute your final grade. We will have 3 group
assignments due the 3rd, 8th, and 15th week of the term. I will hand out details on these as they are assigned;
however, you can expect to hand in a written component (one paper per group) and do a short

presentation of your work in class. Also, there will be one midterm exam and one final exam. Our midterm
will be on Wednesday, October 4th and the final is scheduled for Wednesday, December 13th.
Course Information

Course Information

 Quizzes 50% Group Projects 20% Exams 30%

and the scale

In coordination with the Disability Resource Center, reasonable accommodations will be provided for
qualified students with disabilities. If you need accommodations because of special exceptionalities, please
meet with Brynja Kohler during the first week of the semester to make arrangements. Accommodations
and alternative format print materials (e.g., large print, audio, diskette or Braille) are available through the
Disability Resource Center, located in Taggart Student Center room 104, phone number 797-2444.

Approximate Schedule

 Week 1 Algebra with complex numbers Week 2 Introduction to first order differential equations Week 3 Applications of first order linear equations Nonlinear first order equations Week 4 Matrix algebra, determinants, linear independence Week 5 Introduction to linear systems Week 6 Abstract vector spaces, bases and dimension Week 7 Linear systems, eigenvalues and eigenvectors Week 8-9 Fundamental solution matrices Week 10-11 Second order linear equations Week 12-13 Higher order equations Week 14-15 Laplace transforms

 Week 1 (4) 8/28 0.1-0.3 1.1-1.2 quiz 1 Week 2 (3) 9/4 Labor Day 1.3-1.4 quiz 2 Week 3 (4) 9/11 Group project 1 due 1.5 0.4-0.5 quiz 3 Week 4 (4) 9/18 0.6-0.9 quiz 4 Week 5 (4) 9/25 0.10-0.11 Vector spaces quiz 5 Week 6 (4) 10/2 Vector spaces 10/4 - Midterm 2.1-2.2 Week 7 (4) 10/9 2.3-2.7 quiz 6 Week 8 (3) 10/16 Group project 2 due0 complex numbers 0.1 introduction pp.3-4 #1-14 0.2 cartesian and exponential forms pp.8-9 #1-11 0.3 roots of polynomial equations and numbers p.13 #1-7 1 first-order differential equations 1.1 preliminaries 1.2 definitions pp.62-63 1-3,5 1.3 the first-order linear equation 1.3.1 homogeneous equations p.69 1-9 odd, 12-15 1.3.2 non-homogeneous equations pp.75-76 1-5,8,9,11 1.4 applications of first order linear equations 1.5 non-linear equations of first order 1.5.1 separable equations 0 complex numbers 0.4 matrix notation and terminology pp.16-17 #1-6,8-12 0.5 the solution of simultaneous equations p.22 1,3,5,6,8,10,12,14,15 0.6 the algebra of matrices pp.25-26 1-14 0.7 matrix multiplication pp.30-32 1-19 0.8 the inverse of a matrix pp.36-37 1-6 0.9 the computation of the inverse p.40 2,5,8,11,14,16,17,18-22 0.10 determinantsp.178 1,2,4,7 3.5 the nonhomogeneous equation pp.179-180 1-3 3.6 constant coefficient equations 3.6.1 real and unequal roots 3.6.2 real and equal roots 3.6.3 complex roots pp.187-188 1-31 odd 3.7 spring-mass system in free motion 3.7.1 undamped motion p.192 1-4 3.7.2 damped motion pp.198-200 1,2,4,6-9 3.8 the electric circuit pp.204-205 1-3 3.9 undetermined coefficients pp.211-212 1-33 odd 3.10 the spring-mass system: forced motion 3.10.1 resonance p. 216-217 1-3 3.10.2 near resonance p.221 1-3 3.10.3 forced oscillations with damping 4 higher order equations 4.1 introduction 4.2 the homogeneous equation 4.2.1 general solutions p.237 1-7 4.2.2 initial- value problems pp. 241-242 1-4 4.3 the non-homogeneous equation p.248 1-4, 5-13 odd 4.4 companion systems pp.259-260 1-15 odd, 16 4.5 homogeneous companion systems p.265 1-15 odd 4.6 variation of parameters p.269 1,7,11,15,16 3.1,3.3 10/20 - Fall Break Week 9 (4) 10/23 3.4-3.6 quiz 7 Week 10 (4) 10/30 3.7-3.10 quiz 8 Week 11 (4) 11/6 4.1-4.4 quiz 9 Week 12 (4) 11/13 4.5-4.6 3.2 quiz 10 Week 13 (1/2) 11/20 5.1-5.2 11/22-11/24 - Thanksgiving break Week 14 (4) 11/27 5.3-5.7 quiz 11 Week 15 (4) 12/4 Group project 3 due 5.8-5.10 12/8 - Last class Final 12/130.10.1 definitions and fundamental theorems p.45 1,2,3,5,7,9,11,12 0.10.2 minors and cofactors pp.47-48 2,3,4,5-8 0.11 linear independence pp.53-54 1-10 2 linear systems 2.1 introduction 2.2 eigenvalues and eigenvectors pp.110-111 1-9 odd, 13,16,19,21-38 2.3 first-order systems 2.3.1 complex eigenvalues 2.3.2 repeated eigenvalues pp. 126-127 1-11 odd, 14,15,17,19,20,24 2.4 solution and fundamental solution matrices pp. 135-136 1-7 odd, 12,14,16,21- 25,28 2.6 solutions of nonhomogeneous systems 2.6.1 undetermined coefficients pp.149-150 1,2,6,8,9,14,18,19 2.6.2 variation of parameters pp.154-155 1,3,4,10,12,13,14 2.7 nonhomogeneous initial-value problems p.157 1-4 3 second-order linear equations 3.1 introduction 3.3 linear differential operators pp.170-171 1-8 3.4 linear independence and the wronskian3 second-order linear equations 3.2 sectionally continuous functions p.168 1-3 5 the laplace transform 5.1 introduction 5.2 preliminaries p.279 2,5,8,11,14 5.3 general properties of the laplace transform pp.286-287 1-15 odd, 27 5.4 sectionally continuous functions pp.292-293 1-3,6,8 5.5 laplace transforms of periodic functions p.297 1-5 5.6 the inverse laplace transform p.302 1-15 odd 5.7 partial fractions 5.7.1 nonrepeated linear factors p.309 5,21 5.7.2 repeated linear factors p.313 5,21 5.7.3 repeated quadratic factors 5.8 the convolution theorem p.318 1,4 5.9 the solution of initial-value problems p.323-324 1-15 odd 5.10 the laplace transform of systems p.329-331 5,24,29
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