Math 20     Spring 2003 Introduction to Linear Algebra and Multivariable Calculus

Course Head: John Mackey, Science Center 331, 496-5211, jfm@math.harvard.edu

Introduction and Goals: It is not uncommon to use a list to describe an object. For example, to describe a textbook, one could list its height, width, thickness, number of pages and price. Such a list can be regarded as a vector. In Math 20 we will learn about vectors and see how to analyze them. We will compute their lengths, find the angle between pairs of vectors and do other useful calculations.

We will frequently find it useful to multiply a vector by an object known as a matrix in order to obtain a new vector. Thus, we will study matrices (and their relationship with vectors) in detail, along with computations related to them.

Finally, we will return to the realm of Calculus where we will extend the notions of derivative and maximization to functions which depend on more than one variable. For example, profit may depend on the number of consumers and time, as well as, the number of units produced.

Necessary Background: The prerequisite is as follows: Mathematics 1b or equivalent, or an A or A- in Mathematics 1a, or a 5 on the AB or a 3 or higher on the BC Advanced Placement Examinations in Mathematics.

Classes and Problem Sessions: Math 20 meets mwf at 9am in SC 309. The Course Assistants Tanner Fahl, Keziah Cook and Christina Mulligan,will grade homework and hold weekly problem sessions. I strongly encourage you to attend these problem sessions as they are an integral part of the course and will be devoted primarily to working problems and amplifying the material presented in class. A schedule of problem sessions will be posted on the course website.

Question Center: In addition to class, problem sessions, and office hours, the Mathematics Department operates a Question Center in Loker on Sunday, Monday, Tuesday, Wednesday, and Thursday evenings from 8pm to 10pm. The Question Center will be staffed by Course Assistants from Math 1a, 1b, 21a, and 21b and by graduate students and others. You are encouraged to use this resource as you do your homework and when questions arise.

Homework: Homework exercises are an integral part of the course. It is difficult to understand the material and do well on the exams without working through the homework problems in a thoughtful manner. Please think about the problems posed, your strategies, the meaning of your computations, and the answers you get.

Homework is due at the beginning of the class period following the one in which it has been assigned. Although discussion of the homework with your peers is encouraged, copying any part of another person's homework is not permitted. As a courtesy to the CAs late homework will generally not be accepted. If circumstances cause an assignment to be late, please contact me and I will determine whether to accept the homework. Homework solutions will be posted on the website weekly.

Main Text:  Elementary Linear Algebra, 8th edition by Anton and Rorres. Available at the COOP.

Additional Text:  Chapter 11 of Stewart's Multivariable Calculus, Available at the COOP.

Exams: There will be two midterms and a cumulative final exam. The dates of the midterm exams are as follows:

Exam 1: Wednesday, February 26 from 5:30-7:30 in SC E

Exam 2: Wednesday, April 2 from 5:30-7:30 in SC E

Grading: Your course grade will be determined as follows:

Exam 1: 25%
Exam 2: 25%
Homework: 20%
Final Exam: 30%

Calculators: Calculators will be allowed on the exams only for elementary arithmetic calculations (ie addition, subtraction, multiplication and division).

Week by week schedule (tentative):

Week 1 (January 29 -31):

• Section 1.1 Introduction to Systems of Linear Equations
• Section 1.2 Gaussian Elimination

Week 2 (Feb 3-7):

• Section 1.3 Matrices and Matrix Operations
• Section 1.4 Inverses; Rules of Matrix Arithmetic
• Section 1.5 Elementary Matrices and a Method for Finding A Inverse

Week 3 (Feb 10-14):

• Section 1.6 Further Results on Systems of Equations and Invertibility
• Section 2.1 The Determinant Function
• Section 2.2 Evaluating Determinants by Row Reduction

Week 4 (Feb 19-21):

• Section 2.3 Properties of the Determinant Function
• Section 2.4 Cofactor Expansion; Cramer's Rule

First Mid-Term
Wednesday, February 26 from 5:30 to 7:30 in SC E

Week 5 (Feb 24-28):

• In-class Review for Midterm 1
• Section 3.1 Introduction to Vectors
• Section 3.2 Norm of a Vector; Vector Arithmetic

Week 6 (March 3-7):

• Section 3.3 Dot Product; Projections
• Section 3.4 Cross Product
• Section 3.5 Lines and Planes in 3-Space

Week 7 (March 10-14):

• Section 4.1 Euclidean N-Space
• Section 4.2 Linear Transformations
• Section 4.3 Properties of Linear Transformations

Week 8 (March 17-21):

• Section 7.1 Eigenvalues and Eigenvectors
• Section 7.2 Diagonalization
• Section 11.18 Age-Specific Population Growth

Second Mid-Term
Wednesday, April 2 from 5:30 to 7:30 in SC E

Week 9 (March 31- April 4):

• In-class Review for Midterm 2
• Section 11.? Students select an application
• Section 11.? Students select an application

Week 10 (April 7-11):

• Section 11.? Students select an application
• Section 11.? Students select an application
• Section 11.? Students select an application

Week 11 (April 14-18):

• Section 11.? Students select an application
• Review of Single Variable Calculus
• Multivariable Section 1

Week 12 (April 21-25):

• Multivariable Section 3
• Multivariable Section 4
• Multivariable Section 6

Week 13 (April 28 - May 2):

• Multivariable Section 7
• Multivariable Section 8
• Review of Multivariable Calculus

Reading Period
Saturday, May 3 - Wednesday, May 14

Exam Period
Thursday, May 15 - Friday, May 23