MATHEMATICS 2270-2 Linear Algebra Fall semester 2001

text: when: where:

Linear Algebra with Applications, second edition, by Otto Bretscher MTuWF 12:55-1:45 NS 201

instructor: office: telephone: email: office hours:

Prof. Nick Korevaar

JWB 218 581-7318 korevaar@math.utah.edu M 2-2:50 p.m., T 9-10 a.m., W 2-2:50 p.m., Th 11 a.m - 12:30 p.m.

course web page: www.math.utah.edu/∼korevaar/2270fall01.html

prerequisites: Math 1210-1220, or Math 1250-1260, first year Calculus. Previous exposure to vectors, either in a multivariable Calculus course (e.g. 2210 or 1260) or in a Physics course, is helpful but not essential.

course outline: This is the first semester in a year-long sequence devoted to linear mathematics. Our topic this semester is linear algebra, a fundamental area of mathematics that is used to describe and study a multitude of subjects in science and life. The origins of this field go back to the algebra which one must solve to find the intersection of two lines in a plane, or of several planes in space, or more generally the solution set of one or more simultaneous “linear” equations involving several variables.

We are trying a new text this year which seems more dynamic than the one we used previously. It is our intention to cover chapters 1-8. The detailed syllabus which follows the course summary below is an educated guess at how we will proceed, although the only things for certain are the exam dates.

The course begins in chapter 1 by studying linear systems of equations and the Gauss-Jordan method for systematically solving them. Linear algebra always has a “linear geometric” interpretation, and we begin studying the linear geometry of the Euclidean plane in chapter 2, as well as the relation between inverse matrices and inverse transformations. In chapter 3 we undertake a more systematic exploration of the linear geometry related to transformations and subspaces of ℝ^{n}.

The relatively concrete concepts for subspaces of ℝ^{n} which we discuss in chapter 3, concepts including span, independence, basis, dimension and coordinates, actually apply to many other spaces, called “vector spaces” or “linear spaces”. These generalized notions have many common applications to seemingly diverse areas of mathematics, including the study of differential equations in Math 2280. So, in chapter 4 we study these notions abstractly.

You know what it means for two directions to be perpendicular, and may already have used the “dot product” to test for this condition. This notion of “orthogonality” is a major theme of linear algebra, and is the focus of chapter 5. We will study orthogonal projections and transformations, Gram-Schmidt orthogonalization, methods of least squares, notions of orthogonality for functions, Fourier series.

You have probably used determinants as a computational tool in high school algebra, but are probably not aware of all their uses and why their magic properties work. We will study determinants in detail in chapter 6, including their important geometric meaning related to oriented areas and volumes.

Related to the geometry of linear transformations there are special vectors known as eigenvectors. They also arise in the study of dynamical systems and in differential equations. These are the topics of chapter 7. In chapter 8 we will see some initial applications of eigenvectors, related to conic sections, quadric surfaces, and the multivariable second derivative test. In Math 2280 you will see many more applications.

computer projects: There will be approximately 3 computer projects during the semester, to enhance and expand upon the material in the text. They will be written in the software package MAPLE. On MAPLE days we will meet in the Math Department Computer Lab located in the South Physics building, room 205. This building lies just north of the Math Department building JWB. We do not assume you have had any previous experience with this software and we will make the necessary introductions during the first visit to the lab.

tutoring center: The Math Department Tutoring Center is located in Mines 210, and is open for free tutoring from 8 a.m. to 8 p.m. on M-Th, and from 8 a.m to 2 p.m. on Friday. Some, but not all of the math tutors welcome questions from Math 2270 students. To see the times and specialities of various tutors, consult the web address www.math.utah.edu/ugrad/tutoring.html .

grading: There will be two midterms, a comprehensive final examination, and homework. (Homework assignments and other course information will be posted on the course web page.) Each midterm will count for 20% of your grade, homework Iincluding book and Maple assignments) will count for 30%, and the final exam will make up the remaining 30%. The book homework will be assigned daily and collected weekly, usually on Fridays. Maple projects will generally be due one week after they are assigned. A homework grader will partially grade your assignments. The value of carefully working homework problems is that mathematics (like anything) must be practiced and experienced to be learned.

It is the Math Department policy, and mine as well, to grant any withdrawl request until the University deadline of Friday October 19.

ADA Statement: The American with Disabilities Act requires that reasonable accomodations be provided for students with physical, sensory, cognitive, systemic, learning, and psychiatric disabilities. Please contact me at the beginning of the semester to discuss any such accommodations for the course.

Tentative Daily Schedule

exam dates fixed, daily subject matter approximated

W F

M T W F

M T W F

M T W F

M T W F

M T W F

M T W F

M T W F

M T W F 22 Aug 24 Aug

27 Aug 28 Aug 29 Aug 31 Aug

3 Sept 4 Sept 5 Sept 7 Sept

10 Sept 11 Sept 12 Sept 14 Sept

17 Sept 18 Sept 19 Sept 21 Sept

24 Sept 25 Sept 26 Sept 28 Sept

1 Oct 2 Oct 3 Oct 5 Oct

8 Oct 9 Oct 10 Oct 12 Oct

15 Oct 16 Oct 17 Oct 19 Oct 1.1 1.2

1.3 cont’d 2.1 2.2

none 2.2-2.3 2.3 extra

Maple I 2.4 3.1 3.2

3.2-3.3 3.3 3.4 review

1-3 4.1 4.2 4.3

4.3 5.1 5.1 none

5.2 5.2-5.3 5.3 5.4

Maple II 5.5 5.5 6.1 linear systems matrices and Gauss elimination

solutions of linear systems

cont’d

linear transformations geometric interpretation

Labor Day and inverses inverses fractals

Maple lab

matrix products kernel and image subspaces

independence and bases

dimension coordinates chapters 1-3

exam 1

vector spaces

linear transformations

coordinates

cont’d

orthonormal bases

cont’d

fall break day

Gram-Schmidt

and orthogonal transformations cont’d least squares

Maple lab

inner product spaces

Fourier series

intro to determinants T W F

M T W F

M T W F

M T W F

M T W F

M T W F

M T W F

Th 22 Oct 23 Oct 24 Oct 25 Oct

29 Oct 30 Oct 31 Oct 2 Nov

5 Nov 6 Nov 7 Nov 9 Nov

12 Nov 13 Nov 14 Nov 16 Nov

19 Nov 20 Nov 21 Nov 23 Nov

26 Nov 27 Nov 28 Nov 30 Nov

3 Dec 4 Dec 5 Dec 7 Dec

13 Dec 6.2 6.2 6.3 6.3

review exam 2 7.1 7.1-7.2

7.2-7.3 7.3 7.4 7.4

7.5 7.5 extra extra

7.6 8.1 8.1 none

8.2, Maple III 8.2 extra extra

8.3 8.3 review none

FINAL EXAM determinant properties cont’d geometric meaning cov in multiple integrals

4-6 4-6 dynamical systems and eigenvalues

eigenvectors

cont’d

diagonalization

cont’d

complex numbers and eigenvectors Jordan form Jordan form

stability

symmetric matrices

cont’d

Thanksgiving break

quadratic forms conics and quartics second derivative test cont’d

singular values

cont’d

review for final U. reading day

entire course 1:00-3:00 p.m.