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Linear Algebra - Math 304

Spring 2020 - Course Coordinator: Prof. Alexander Borisov

1Alexander BorisovWH-109777-2764borisovMWF:8:00-9:30SL-302
2Casey DonovenWH-202777-2982cdonovenMWF:8:00-9:30WH-G002
3David BiddleWH-126 biddleMWF:9:40-11:10AA-G021
4Charles (Matt) EvansWH-380 evansMWF:11:20-12:50LH-12
5David BiddleWH-126 biddleMWF:11:20-12:50AA-G021
6Alex FeingoldWH-115777-2465alexMWF:1:10-2:40SL-302
7Thomas KilcoyneWH-336 kilcoyneMWF:2:50-4:20LH-12
8Thomas KilcoyneWH-336 kilcoyneMWF:4:40-6:10LH-12

(*): Each email address in this table is of the form but that should happen automatically if you just click on the link.

If a section has its own detailed syllabus webpage, a link to that page will be provided under the Instructor column of the table above.

Below is a partial syllabus with information for all sections that you should know. Your instructor may have a more detailed syllabus about how your section will be run.

Important: Syllabus Adjustments due to the Coronavirus Epidemic

Syllabus Addition for MATH 304, due to the Online Instruction Mode

The following changes will be implemented in all sections of MATH 304, effective 03/19/2020.

1) Instead of the regular classes, the instructors will be holding online office hours, using Zoom or alternative software. These online office hours will be held during the regular instructional times. Additional online office hours may be scheduled as needed. In addition to the recommended textbook reading, there will be links to videos posted online. You are responsible for learning the material on your own, and using the online office hours to clarify any questions that you may have.

2) Instead of the in-class quizzes you will be getting homework assignments, through WebWork, MyCourses, or other system, at your instructor's discretion. These assignments are on top of the regular WebWork homework, and they will differ by section and may have shorter deadlines.

3) Midterm examinations and final examination will be conducted in a various remote formats, depending on the section. Your instructor will provide you with the details.

4) Since we are unable to hold a common in-class final examination, the one-letter-grade rule from the original syllabus will not apply.

5) It is your responsibility to inform your instructor of any circumstances that hinder or jeopardize your participation in the course, including any computer-related issues. The instructors and the course coordinator will do their best to help you learn and have a successful semester.

Consider the current situation, as unfortunate as it is, as an opportunity to develop a valuable skill of learning independently. It is, actually, one of the main goals of college education, much more important than any specific knowledge in any course.


``Linear Algebra” by Jim Hefferon, Third Edition, available as a free download here:

Linear Algebra by Jim Hefferon.

Students can also buy a cheap printed version from the link on the author's website.

Here are also some additional books that students and instructors may find helpful.

A First Course in Linear Algebra by Robert A. Beezer

Elementary Linear Algebra by K.R. Matthews

Linear Algebra by D. Cherney, T. Denton, R. Thomas, and A. Waldron

Exam Schedule

Only the Final Exam will be common to all sections, but we expect each section will administer other exams on the same day, three times during the semester, as follows:

Exam 1: February 17 or 19 (depending on section)

Exam 2: March 18 This Exam will have to be given online since all classes March 17, 18 have been cancelled.

Exam 3: April 22 or April 24, depending on your section. This Exam will have to be given online.

Final Exam: Thursday, May 7, 8:00-10:00 AM, will be given online.

Arrangements for online exams will be announced soon. It is your instructor's responsibility to find a way to administer such exams, get the results and return the graded exam results.

Grade Distribution and Policies

The course total will be determined as follows:

Homework: 5%

Quizzes: 20% (the number and scope to be determined by your instructor)

Exam 1, 2, and 3: 15% each

Final Exam: 30%

The general grade cutoffs are going to be the following:

90% A; 80% B; 70% C; 60% D; and proportional cutoffs for A-,B+,B-,C+, and C-.

These cutoffs may be relaxed at the end of the semester, and may also differ a bit from section to section.

This rule from the original syllabus will not apply: (Additionally, the following one-letter-grade rule will apply: the grade in the course cannot exceed the grade on the final examination by more than one letter grade. For example, if you get a grade of B- on the final, the highest grade you can get in the course is A-. Note that the rule only works one way: even if you get an A on the final, you may still, theoretically, fail the course. The purpose of this rule is to bring some uniformity to the grading, considering that each section will have their own exams, quizzes, and cutoffs.)


Online homework will be done using WebWork. The server address is

Your WebWork account username is the pre@ portion of your e-mail account. Your initial password is the same as the username. For example, if your Binghamton e-mail account is then your username: xyzw77 and your password: xyzw77

Make sure to change your password as soon as possible to a secure password, and save that choice where it will not be lost.

Besides the online homework sets, you should do problems from the book, as selected by your instructor, see an approximate schedule below. This part of the homework will not be graded, but it will be paramount to your success in the course.

Expected workload outside of the classroom

This class is scheduled to meet three times per week for 90 minutes each time. In addition to attending all classes, you should expect to need 8 to 10 hours per week outside of the class meetings to study the material and do homework.

Expected behavior in class

During classes all students are expected to behave according to university rules. Your instructor makes the final decisions about what to allow in the classroom, regarding in particular cell phone and laptop use, and food and beverage consumption. If you have any temporary of permanent needs that may necessitate an exception, it is your responsibility to discuss the matter with your instructor in advance.

Disability Information

If you have a disability for which you are or may be requesting an accommodation, please contact both your instructor and the Services for Students with Disabilities office (119 University Union, 607-777-2686) as early in the term as possible.


January 22 First Day of Classes: After meeting your instructor you should have been given this webpage address to check for the syllabus and instructions. Step 1: Read the entire syllabus, including the detailed syllabus for your section. Step 2: Set up your WebWork account for doing homework assignments. Step 3: Attend all classes and keep up with all homework corresponding to your section lectures.

February 11: Prof. Borisov will hold a review for all MATH 304 students on Sunday, Feb. 16, at 1:00-3:00 pm, in LH-14. The review session will be based entirely on student's questions, the posted sample exams or any other sources.

March 16: Today the administration announced that all classes (in person and online) have been cancelled for March 17 and 18, so that online only classes will begin March 19. That prevents having the scheduled March 18 Exam 2 in class. Each instructor will tell their students how Exam 2 will be handled.

March 30: Lectures of Prof. Feingold (Section 6) recorded on Panopto as well as written lecture notes are available through links on his webpage: Feingold Section 6 Webpage. These recordings are 90-minute class lectures presented online including interactions with students, and the written lecture notes are copies of the pages shown during the lectures.

Tentative Schedule

Unless otherwise specified, the Text is the Jim Hefferon's book and the exercises are from there, in the format “Chapter:Section.Subsection.ProblemNumber”. It is subject to change and adjustment at your instructor's discretion. NOTE: The Problems are for practice only and are not to be turned in. There will be separate weekly GRADED HOMEWORK through WebWork. Many of the examination problems will be similar to these practice problems and/or the WebWork problems.

Week Dates Topics Text Problems
1 Jan 22, 24 Introduction, preview, examples; linear combination Ch. 1, I.1 1:I.1.17,1.19,1.21
Gaussian elimination (reduction) Ch. 1, I.1 1:I.1.22,24,27,32
2 Jan 27-31 (Augmented) matrix of a system, solution set Ch. 1, I.2 1:I.2.15,16,17,18,20,23
Logical statements, basic constructions, quantifiers A-1, A-2
Induction (informal), sets, functions A-3, A-4
3 Feb 3-7 Homogeneous and non-homogeneous systems (no formal induction in Lemma 3.6) Ch. 1, I.3 1:I.3.15,17,18,20,21,24
Points, vectors, lines, planes Ch. 1, II.1 1:II.1.1,2,3,4,7
Distance, dot product, angles, Cauchy-Schwarz and Triangle Inequalities Ch. 1, II.2 1:II.2.11,12,14,16,17,21,22
4 Feb 10-14 Gauss-Jordan reduction, reduced row echelon form Ch. 1, III.1 1:III.1.8,9,10,12,13,14,15
Linear combination lemma, uniqueness of RREF (skip proofs of 2.5, 2.6) Ch. 1, III.2 1:III.2.11,14,20,21,24
Review for Examination 1 Ch. 1; Appendix Sample_1,Answers_1; Sample_2,Answers_2; Sample_3,Answers_3
5 Feb 17-21 Examination 1 Jan 22 – Feb 14
Matrix operations, including the transpose. Linear system as a matrix equation Matthews 2.1 3:III.1.12,13,14,15
Linear maps (transformations) given by matrices Matthews 2.2 3:III.1.18; 3:III.2.17,30
6 Feb 24-28 Vector spaces: definition, examples Ch. 2, I.1 2:I.1.17,18,19,21,22,29,30
Subspaces. Span Ch. 2, I.2 2:I.2.20,21,23,25,29,44
Linear independence Ch. 2, II.1 2:II.1.20,21,24,27
7 Mar 2,4 Properties of linear independence Ch. 2, II.1 2:II.1.28,29,31,32
Basis of a vector space Ch. 2, III.1 2:III.1.18,19,20,21,22,23,24,28,29,32
Mar 6 No class Winter Break
8 Mar 9-13 Dimension of a vector space Ch. 2, III.2 2:III.2.16,17,18,19,22,23,26,29
Column space, row space, rank Ch. 2, III.3 2:III.3.17,18,19,20,21,23,29,32,39
Range space and null space Ch. 3, II.2 3:II.2.21,23,24,26,31,35
9 Mar 16-20 Review for Examination 2 Sample_1,Answers_1; Sample_2,Answers_2; Some_Practice_Problems, Answers
Examination 2 Feb 19 – Mar 16
Invertible matrices: definition, equivalent conditions; inverse matrix Ch.3, IV.4 3:IV.4.13,14,15,16,17,18,19,26,29 InvertibleMatrices_1 InvertibleMatrices_2 InvertibleMatrices_3 InvertibleMatrices_4 InvertibleMatrices_5
10 Mar 23-27 Elementary matrices. Row reduction using elementary matrices Ch. 3, IV.3; CDTW Ch. 2, 2.3 3:IV.3.24,25,34; ElementaryMatrices_1 ElementaryMatrices_2 ElementaryMatrices_3
Determinant of a matrix, properties Ch. 4, I.1, I.2 4:I.1.1,3,4,6,9; 4:I.2.8,9,12,13,15,18 Determinants_1 Determinants_2 Determinants_3 Determinants_4 Determinants_5 Determinants_6
More on Determinants Ch. 4, II.1, III.1 4:III.1.11,14,16,17,20,21,22 Determinants_7(Cramer) Determinants_8(Adjoint)
11 Mar 30-Apr 3 Linear maps (transformations) between general vector spacesCh. 3, II.1 3:II.1.18,19,20,25,26,28
Matrix of a linear transformation, matrix of the composition, inverseCh. 3, III.1, IV.2 3:III.1.12,14,15,18,19,20,21,22,26,29 Matrix_of_Transformation_1
Change of basis, similar matrices Ch. 3, V.1, V.2; Ch. 4, I.1 3:V.1.7,9,10,11; 4:I.1.5,8,11,13,14 Matrix_of_Transformation_2 Matrix_of_Transformation_3 Matrix_of_Transformation_4 Similar_Matrices
Apr 4-12 No Class Spring Break
12 Apr 13-17 Complex numbers Matthews 5.1–5.6 Matthews 5.8.1,2,5,6,7,9 Complex_Numbers_1 Complex_Numbers_2 Complex_Numbers_3 Complex_Numbers_4 Complex_Numbers_5
Eigenvectors, eigenvalues, eigenspaces for matrices and linear operators. Characteristic polynomial Matthews 6.1, 6.2; Ch. 5, II.35:II.3.23,26,27,28,29,30,33 Eigenvectors_1 Eigenvectors_2 Eigenvectors_3 Eigenvectors_4 Eigenvectors_5
Diagonalization of matrices Ch. 5, II.2, II.3 5:II.3.21,32,35,40,41,44 Diagonalization_1 Diagonalization_2 Diagonalization_3 Diagonalization_4 Diagonalization_5 Diagonalization_6
13 Apr 20-24 Review for Examination 3 Sample_1, Answers_1; Sample_2, Answers_2; Sample_3, Answers_3
Examination 3 Mar 20 – Apr 20
Orthogonal and orthonormal bases of $R^n$ and its subspaces; orthogonal matrices Ch. 3, VI.1, VI.2 3:VI.1.6,7,17,19; 3:VI.2,10,11,12; Orthogonal_1 Orthogonal_2 Orthogonal_3 Orthogonal_4
14 Apr 27-May 1 Orthogonal complement of a subspace, orthogonal projectionCh. 3, VI.3 3:VI.3.11,12,13,14,26,27 Complements_1 Complements_2 Complements_3 Complements_4 Complements_5
Gram-Schmidt process; orthogonal diagonalization of matricesCh. 3, VI.2 3:VI.2,13,15,17,18,19,22; GramSchmidt_1 GramSchmidt_2 OrthogonalDiagonalization_1 OrthogonalDiagonalization_2
Review for the Final Examination
15 May 4 Review for the Final Examination Sample exams are adapted, with permission, from the collection of Dr. Inna Sysoeva Sample_1, Answers_1; Sample_2, Answers_2; Sample_3, Answers_3; Sample_4, Answers_4; Sample_5, Answers_5
Thu. May 7 8:00-10:00 am GW 69EX Cumulative Final Exam


Syllabi from previous semesters

The syllabus for Math 304 in Fall 2019 is available through this link:

Fall 2019 page

The syllabus for Math 304 in Spring 2019 is available through this link:

Math 304 Syllabus for Spring 2019

The syllabus for Math 304 in Fall 2018 is available through this link:

Fall 2018 page

math304/start.txt · Last modified: 2020/05/01 12:18 by borisov