MATH 235,   LINEAR   ALGEBRA

July 2 - August 10, 2001

Mon, Tues, Wed, Thur   5:45-8:00

Hylan 102

Prerequisites: none

Textbook: Linear Algebra and Its Applications, second edition, by David C. Lay

Topics to be covered:
Systems of linear equations.
Finite-dimensional vector spaces over R (real numbers) and C (complex numbers) presented from two points of view: axiomatically and with coordinate calculations.
Linear transformations, matrices, rank, determinant, eigenvalues, eigenspaces.
Examples and applications.

Schedule       Homework assignments       Projects

Grading policy:

Attendance is compulsory in the sense that you may lose points for classes missed without a good reason such as illness or family emergency. If you are going to miss a class, please send me an email, preferably in advance. The number of points you lose for missed classes depends on how well you do on exams. If you get perfect scores on exams, you won't lose any points for missing classes. The worse you do on exams, the more points you lose for missing classes. The maximum penalty for one missed class is 5 points.

Instructor: Maria Voloshina
Office: Hylan 818
Office hours: M,T,W,R 5-5:40 PM, and by appointment
Email: voloshin@math.rochester.edu
Webpage: http://www.math.rochester.edu/u/voloshin/
Webpage for Math 235: http://www.math.rochester.edu:8080/u/voloshin/math235-01.html


Schedule

July 2, 3, 5, 9
Chapter 1. Linear Equations in Linear Algebra
(Systems of linear equations. Row reduction and echelon forms. Vector equations. The matrix equation Ax=b. Solution set of linear systems. Linear independence. Introduction to linear transformations. The matrix of a linear transformation)

July 10, 11, 12
Chapter 2. Matrix Algebra
(Matrix Operations. The inverse of a Matrix. Characterizations of invertible matrices. Partitioned Matrices. Subspaces of R^n) Review

July 16, 17
Chapter 3. Determinants
(Introduction to determinants. Properties of determinants. Cramer's Rule, Volume, and Linear transformations) Exercises. Practice midterm.

July 18, 19, 23, 24
Chapter 4. Vector Spaces
(Vector spaces and subspaces. Null spaces, Column spaces, and Linear transformations. Linearly independent sets. Bases. Coordinate systems. The dimention of a vector space. Rank. Change of basis)
Midterm (July 24)

July 25, 26, 30
Chapter 5. Eigenvalues and Eigenvectors
(Eigenvectors and eigenvalues. The characteristic equation. Diagonalization. Eigenvectors and linear transformations. Complex eigenvalues) Review

July 31, August 1, 2
Chapter 6. Orthogonality and Least-Squares
(Inner product, length, and orthogonality. Orthogonal sets. Orthogonal projections. The Gram-Schmidt process. Least-Squares problems. Inner product spaces)

August 6
Chapter 7. Symmetric Matrices and Quadratic Forms
(Diagonalization of symmetric matrices. Quadratic forms)

August 7, 8
Presentations of projects. Applications.

August 9
Exercises. Review. Practice Final.

August 10
Final examination   Time to be decided. (We will choose a time that is convenient to all of us)


Homework assignments

These problems should be submitted in written, and will be corrected and graded. Please start working on your assignments as early as possible, so that if you need help you have enough time to get it.

Each problem is worth 1 point. A score of 105 will be considered excellent. There are more than 105 problems here, and you are very welcome, even strongly recommended, to do as many problems as possible. If you do more than 105 problems, consider it as getting some extra credit.

In addition to these problems, you are strongly encouraged to do other problems from your book. The more problems you do, the better you will do on exams. If you want me to look at any other problems that you did, and tell you whether your solutions are correct, I will be happy to do so (however, you will not get credit for those problems).

Assignment 1, due July 9
1.1 (p.10) #1, 9, 21
1.2 (p.25) #5a, 7, 17, 25
1.3 (p.37) #5, 11, 21
1.4 (p.46) #3, 7, 11, 17, 25
1.5 (p.55) #1, 9, 17, 22
1.6 (p.64) # 5, 13, 21, 29, 33

Assignment 2, due July 16
1.7 (p.74) #1, 5, 11, 17, 21
1.8 (p.83) #1, 6, 18
2.1 (p.107) #1, 10, 20, 25
2.2 (p.117) #4, 13, 22, 33
2.3 (p.123) #2, 4, 23, 30, 33
2.4 (p.130) #3, 5, 13
2.9 (p.174) #2, 5, 16, 21, 29
Assignment 3, due July 23
3.1 (p.185) #6, 9, 18, 22, 37
3.2 (p.193) #5, 21, 39
3.3 (p.204) #2, 7, 22, 24
4.1 (p.217) #6, 8, 11, 15, 21
4.2 (p.228) #3, 9, 19, 23, 26
4.3 (p.237) #3, 15, 33
4.4 (p.248) #1, 6, 9, 15, 27
Assignment 4, due July 30
4.5 (p.255) #7, 11, 22
4.6 (p.263) #6, 13, 19, 28
4.7 (p.270) #6, 9, 14
5.1 (p.303) #1, 7, 12, 18, 24
5.2 (p.311) #5, 10, 21, 24
5.3 (p.319) #1, 7, 11, 22, 27
5.4 (p.327) #2, 6, 11, 13, 20
Assignment 5, due August 6
5.5 (p.335) #3, 7, 14, 22
6.1 (p.376) #9, 13, 24, 31
6.2 (p.387) #1, 10, 13, 17
6.3 (p.395) #4, 15, 21
6.4 (p.403) #1, 7, 14, 19
6.5 (p.411) #2, 7, 11
6.7 (p.430) #1, 4, 9, 16, 21
Assignment 6, due August 10
7.1 (p.448) #3, 4, 13, 27, 28
7.2 (p.457) #3, 9, 21






Projects

Everybody must choose one project. All of you should work on different projects, so I suggest that you look at all the projects, and decide with your classmates who is taking which project. Perhaps the best way to do this is to use the course newsgroup, ur.mth235. Tell me your decision by July 16. Nine projects are given below. More projects may be added if needed. The projects are ordered randomly. It is not true that the latter profects are more difficult, so you are encouraged to look at all of them. If you would like to discuss your project with me or anybody else, you are welcome to do so. However, you are expected to write down everything you've done, and give a presentation in class.

Some projecs require using a computer (mostly, in order to multiply large matrices or solve systems of linear equations with many equations and many variables). I recommend you to do this in a computer lab, for example, in Harkness 114 (open Monday thru Friday 8 AM - 5 PM). Click here for a Mathematica primer.

Project I: Applications to Physics (temperatures)
1.1 #35 and 36       1.9 #14       2.5 #31

Project II: Applications to Chemistry
1.5 #39 and 40
Find out and tell us about any other applications of Linear Algebra to Chemistry.

Project III: Determinants

Project IV: Interpolations

Project V: Applications to Calculus
4.1 #35       4.3 #33 and 34       4.5 #34       4.7 #17 and 18

Project VI: Applications to Economics
1.3 #27       1.5 #37 and 38       1.9 #12       4.8 #24

Project VII: Applications to Probability theory

Project VIII: Linear Transformations

Project IX: Applications to Physics (circuits)
Choose any 5 problems from the following list. There are enough problems for 2 people, so if 2 people would like to work on this project, you are welcome to do so, but then please choose different problems.
1.9 #5, 6, 7, 8       2.5 #27, 28, 30       4.3 #28       5.7 #19, 20, 21, 22

Projects X through XIII: Read the given section and do any 5 problems from that section. Try to choose different problems, not similar ones. Both in your paper and during your presentation in class, state the main ideas of the section and show the solutions of the problems you have chosen.

Project X: Section 2.7 "The Leontief Input-Output Model"

Project XI: Section 2.8 "Applications to Computer Graphics"

Project XII: Section 5.6 "Discrete Dynamical Systems"

Project XIII: Section 5.7 "Applications to Differential Equations"
(choose any problems except 19-22 because these are given in project IX)

If you wish to get some extra credit, here are some possibilities:


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This page was last modified on July 3, 2001