MATH 304 SPRING 2008 SYLLABUS

Contact Information

Office: LN-2218, Phone: 777-2465, Email: alex@math.binghamton.edu, Office Hours: MWF 1:10 - 2:10 and by appointment.

Class meets: MWF 12:00 - 1:00 in SW - 321, Th 11:40 - 1:05 in LN - 1120.

Textbook and Course Contents

``Linear Algebra'' by Matthew Brin, published by the Department of Mathematical Sciences, Binghamton University, and available throught the University Bookstore.

The entire book, Chapters 1 - 7, will be covered. A list of major topics which will be covered is as follows:

Systems of Linear Equations

Solution by row reduction

Matrices and operations with them

Reduced Row Echelon Form

Sets of Matrices of size mxn

Functions (general theory), injective, surjective, bijective, invertible, composition

Linear Functions determined by a matrix

Abstract Vector spaces

Basic theorems, examples, subspaces, linear combinations, span of a set of vectors

Linear functions L : V ---> W between vector spaces

Kernel and Range of a linear function

Connection with injective, surjective, bijective, invertible, composition of linear functions, isomorphism

Matrix multiplication from composition, formulas, properties (associativity)

Row (and column) operations achieved by left (right) matrix multiplication by elementary matrices

Theorems about invertibility of a square matrix (if row reduces to the identity matrix), algorithm to compute inverse

Linear independence/dependence, removing redundant vectors from a list keeping span the same

Basis (independent spanning set), Theorems about basis

Dimension of a vector space (or subspace), rank, nullity, theorems about dimension

For L : V ---> W, dim(V) = dim(Ker(L)) + dim(Range(L)) and its applications

Coordinates as an isomorphism from V to nx1 matrices

Representing a linear function L : V ---> W by a matrix (with respect to choice of basis S of V and basis T of W)

Theorems and algorithms, how matrix representing L changes when bases change to S' and T'

Equivalence of matrices, Block Identity Form

Study special case of L : V ---> V using same basis S on both ends, linear operators

Effect of change of basis on matrix representing a linear operator, similarity of matrices

Investigate when L might be represented by a diagonal matrix

Eigenvectors, Eigenvalues

Determinants as a tool for finding eigenvalues, general theorems and properties about determinants

det(AB) = det(A) det(B), A invertible iff det(A) not zero

Characteristic polynomial of a matrix, det(A - t I), roots are eigenvalues

Similar matrices have same characteristic polynomial

Geometric and algebraic multiplicities of eigenvalues for L : V ---> V (or for matrix A representing L)

Theorems (geometric mult less than or equal to algebraic mult), L diagonalizable iff geom mult = alg mult for all eigenvalues

Computational techniques to find a basis of eigenvectors, diagonalization of matrix A, if possible

Extra topics if time allows (usually not enough time for these in the elementary linear course) :

Geometry in Linear Algebra: dot product, angles and lengths, orthogonality, orthonormal sets, orthogonal projections

Orthogonal matrices

Quadratic forms and associated bilinear forms on a vector space V

Matrix representing a bilinear form with respect to a choice of basis S

Effect of change of basis on matrix representing a bilinear form

Classification of quadratic forms

Exams

Information about the scheduling of exams will be posted here when it becomes available.

Exam 1: Feb. 28, 2008.

Exam 2: April 3, 2008.

Exam 3: May 1, 2008.

Final Exam: Monday, May 12, 2008, 8:30 - 10:30 AM, LH - 001.

Anyone with a special problem or a finals conflict must contact the professor as soon as possible to make arrangements. The Final Exam is comprehensive, covering the whole course.

There will be 3 ``hourly'' exams (actually 1 hour and 25 minutes) and 1 Final Exam (2 hours long) during the scheduled Finals period. The hourlies will be worth 100 points each, and the (2-hour) Final Exam will be worth 150 points. The contents of each exam will be determined one week before the exam. The Final Exam will be comprehensive, covering the whole course. ANYONE UNABLE TO TAKE AN EXAM SHOULD CONTACT THE PROFESSOR AHEAD OF TIME TO EXPLAIN THE REASON. A MESSAGE CAN BE LEFT AT THE MATH DEPT OFFICE (777-2147) OR ON PROFESSOR FEINGOLD'S VOICEMAIL (777-2465). NO ONE SHOULD MISS THE FINAL!

Practice Exams

Practice exams and solutions are posted here.

A Practice Exam 1 and its solutions may be found at the following link: Practice Exam 1 and solutions.

A Practice Exam 2 and its solutions may be found at the following link: Practice Exam 2 and solutions.

A Practice Exam 3 and its solutions may be found at the following link: Practice Exam 3 and solutions.

Exams From This Semester and Their Solutions

After exams have been graded and returned, copies of the questions and their solutions will be posted here. This will be very helpful to study for the final exam, and to correct your mistakes.

Exam 1 and its solutions may be found at the following link: Exam 1 and solutions.

Exam 2 and its solutions may be found at the following link: Exam 2 and solutions.

Exam 3 and its solutions may be found at the following link: Exam 3 and solutions.

Grading

Each exam will be curved, giving each student a letter grade as well as a number grade, and the Total of all points earned will also be curved. The letter grades on the exams indicate how a student is doing, and will be taken into consideration in making the curve for the Totals. The course grade will be determined by the curve of Total points earned. Only borderline cases will be subject to further adjustment based on Homework. Any cases of cheating will be subject to investigation by the Academic Honesty Committee of Harpur College.

Homework

For each section of material covered there will be an assignment of problems from the textbook. They will be due one week from the day they are assigned (or the next scheduled class meeting after that if there is a holiday). Late assignments will be accepted at the discretion of the Professor. Assignments will be examined by a grader, who will record the fact that an assignment was attempted, and give some feedback on how selected problems were done. MOST QUESTIONS ABOUT PROBLEMS SHOULD BE ASKED OF THE PROFESSOR AT THE BEGINNING OF CLASS. DO NOT DEPEND ON THE GRADER TO FIND AND CORRECT YOUR MISTAKES. The number of homeworks attempted will be considered as a factor in determining your course grade if you are a borderline case in the Total curve.

Homework Assignments

Homework 1: Exercises (1) - (4), Due Feb. 6.

Homework 2: Exercises (5) - (7), Due Feb. 7.

Homework 3: Exercises (9), Due Feb. 11.

Homework 4: Exercises (10), Due Feb. 15.

Homework 5: Exercises (11), (12), Due Feb. 22.

Homework 6: Exercises (14), (15), (16), Due Feb. 27.

Homework 7: Exercises (17), (19), (20), (21), Due Feb. 28.

Homework 8: Exercises (22), Due Feb. 29.

Homework 9: Exercises (23), (24), (25), (26), (27), Due Mar. 7.

Homework 10: Exercise (34), Due March 10.

Homework 11: Exercises (35), Due March 12.

Homework 12: Exercises (36), (37), (38), Due March 13.

Homework 13: Exercises (39), Due March 17,

Homework 14: Exercises (40), (41), (42), Due March 20.

Homework 15: Exercises (43), (45), (46), (49), (50), (51), (52), Due March 31 because of spring break).

Homework 16: Exercises (54), Due April 11.

Homework 17: Exercises (55), (59), Due April 17.

Homework 18: Exercises (56), (58), Due April 23.

Homework 19: Exercises (60), Due April 24.

Homework 20: Exercises (61), Due April 30.

Homework 21: Exercises (62), Due May 1.

Homework 22: Exercises (64), Due May 7.

Homework 23: Exercises (65), (66), (67), Due May 9.

Homework 24: Exercises (68), (69), Due May 9 Last Day of Classes, last day to hand in any homework.

General Comments

Class attendance is required at both the lectures and the discussion sessions, and sleeping in class does not count as being there. Lectures can be interrupted at any time for questions. At the start of each class be ready to ask questions about homework problems or about the previous lecture. I want to create an atmosphere where you all feel very free to ask questions and make comments. If anyone feels that I have not answered a question clearly, completely, and with respect and consideration for the student who asked it, I want you to let me know about it immediately so I can correct the problem. You can do this in class or in my office hours, verbally or in writing, on paper or by email, or by whatever means makes you most comfortable, but with enough detail that I understand what you think I did wrong. It will be too late to help if you only tell me at the end of the course when I do a Student Opinion of Teaching survey.

The material is a combination of theory and calculation, and it is necessary to understand the theory in order to do sensible calculations and interpret them correctly. There is a significant difference between training and education, and I feel strongly that our goal at this university is to educate you, not just to train you to do computations. Theory is not presented to impress you with my knowledge of the subject, but to give you the depth of understanding expected of an adult with a university education in this subject. I will try to give you the benefit of my 37 years of experience teaching mathematics at the university level, but it will require your consistent concentrated study to master this material. While much learning can take place in the classroom, a significant part of it must be done by you outside of class. Using the book, class notes, homework exercises, only you can achieve success in this course. Students who do not take this course seriously, who do not take the advice I give, are not likely to be rewarded at the end. I am here to help and guide you, and I also make and grade the exams to judge how much you have learned, but grades are earned by you, not given by me. Exams will be a combination of theory questions and calculations appropriate for a course of this level.