Math 34600: Elements of Linear Algebra

Supervisor: Matthew Auth

Vector spaces, basis and dimension, matrices, linear transformations, determinants, solution of systems of linear equations, eigenvalues, and eigenvectors.

Pre- or co-requisite: Math 21300 or departmental permission. 3 HR./WK.; 3 CR.

Syllabus, Class Schedule, and CLO

Syllabus, Class Schedule, and CLO

Grading Factors

HW Average: 5% of course grade.

Quiz Average: 15% of course grade.

Midterm Exam 1: 20% of course grade.

Midterm Exam 2: 20% of course grade.

Final exam 40% of course grade

There will be no make-up quizzes. If you miss one of our two in-class exams, your final exam will serve as your make-up in-class exam. All topics covered on the in-class exams will be covered on the final exam. There will be a make-up final exam if you are ill on the final exam day.

Moreover, at the end of the course if your final exam average is superior to your in-class exam 1 grade or your in-class exam 2 grade, then you can use your final exam grade to replace all lower in-class exam average(s). For instance, if your hw grade is 100, quiz average is 85, midterm 1 grade 86, midterm 2 grade 77, and final exam grade 79 then your course grade will be computed as 5% * 100 + 15% * 85 + 20% * 86 + 60% * 79 = 82.35 course grade, which would be a B-.

In-Class Exam Dates

EXAM 1: All Monday/Wednesday sections will take exam 1 in class on Tuesday 10/13. All Tuesday/Thursday sections will take exam 1 in class on Tuesday 10/6.

EXAM 2: All Monday/Wednesday sections will take exam 2 in class on Monday 11/23. All Tuesday/Thursday sections will take exam 2 in class on Tuesday 11/24.

Textbook

Linear Algebra with Application by Otto Bretscher, 5th Edition Pearson. Several copies of this text are on hold in the Marshak science library. Linear algebra is very different from your previous math courses. In prior math classes the difficult computations used in the class were the most challenging part of the course. In linear algebra the concepts are as important as the computations. In this basic linear algebra course the computations will be mostly simple. It will be challenging, however, to decide which computational technique to use in a given problem. In order to do this you must learn how the concepts and ideas are related. This takes time and effort, and it is uncomfortable to do at first. Most students must read each section several times before the new concepts and terminology begin to make sense. The true/false questions at the end of each chapter force students to think conceptually about the material in the course. Please work through many of these true/false questions. It doesn't help to guess. You should be able to justify each true or false answer.

Practicing Flipped Classroom

Here is some good advice: Read assigned sections from our textbook before class. Learning how to read a math textbook will be essential in this course as well as in all your future math courses. Read the section you will cover in class before coming to class. You will not fully understand the topic after a first reading. Do not worry. You will pick up enough of the idea to make your time in class more productive. After class you will then most likely need to reread the section multiple times to become comfortable with the material.

Exercise Sets (Homework)

Homework sets will be due weekly on Sunday nights in Brightspace.

The best way to learn the material is to consistently do many suggested textbook exercises. It is better to spend a couple hours every day doing linear algebra problems rather than spending the weekend before the exam cramming for eight hours a day. Consider the final exam like running a marathon and the problems as your training sessions. If you consistently do problems (train) then you will be well-prepared for the final exam. However if you have long gaps between working on problems you will always have difficulty getting back into the math mindset necessary to complete the problems. The first day of training is always the most painful.

Quiz and exam questions will be similar to the assigned textbook homework questions, exact replicas sometimes. You should attempt all the assigned homework problems to learn the material and prepare for quizzes and exams. If you have done all assigned hw problems, there should be no surprises on quizzes and exams.

Justifying an answer to a True/False question is an informal proof. Justifying a statement is false can often be done by providing a counter-example. For instance, the statement "All 4x5 matrices of rank 4 have nullity zero" is a false statement. In order to justify the statement is false you could write a particular matrix of your choice (use a row reduced one) with 4 rows and 5 columns of rank 4 that has a nonzero vector in its kernel. However the similar statement "All 5x4 matrices of rank 4 have nullity zero" is a true statement. Justifying true statements in linear algebra often takes more work. A particular matrix cannot be used to verify that this statement is true for all 5x4 matrices. Instead you can verify the statement by writing something like, "in all 5x4 matrices of rank 4, each variable is a leading one and there are no free variables and therefore no nonzero vectors in the kernel."

Later in your career at CCNY and beyond linear computations will become more difficult, often because the matrices will be huge. The computational techniques used in such problems will be similar to those we use in this course but a single problem will contain many more basic computations than a single human can do reliably. The core ideas and concepts you learn in this course remain be essential. In such problems you will have a computer make the computations. Modern computers have been fine-tuned to solve large linear algebra problems quickly and reliably.

Topic Summaries, Videos, Practice Problems

Topic Summaries, Videos, Practice Problems

Getting Help + Problem Sessions

Final piece of good advice: Form study groups and go to office hours or tutoring when you inevitably get stuck. We will be holding math 346 problems sessions Tuesdays and Fridays 12:30-1:30 where you will work on 346 problems in groups with a tutor.

Do not spend more than a day or two being stuck on a problem or try to master a concept from class or our textbook. There is not enough time in the semester to remain stuck and not practicing.

Artino Math Tutoring

Marshak Tutoring

Sample Final Exams

Final exam questions will be modeled on the suggested textbook exercises, sometimes the quiz and exam questions are identical to suggested textbook exercises. If you've worked through a wide portion the suggested textbook exercises, there should be no surprises on exams.

Sample Final Exam A

Sample Final Exam B

Sample Final Exam C

Sample Final Exam D

Sample In-Class Exams

All quiz and exam questions will be modeled on the suggested textbook exercises, sometimes the quiz and exam questions are identical to suggested textbook exercises. If you've worked through a wide portion the suggested textbook exercises, there should be no surprises on exams.

Sample Exam 1A (section 1.1 up to and including 3.3 on syllabus)

Sample Exam 1B. Another sample exam 1.

Sample Exam 1C. Another sample exam 1.

Sample Exam 2A. (sections 3.4 up to and including 7.1 on syllabus)

Sample Exam 2B. Another sample exam 2

Sample Exam 2C. Another sample exam 2

Sections

For Fall 2026, the following sections are being offered:

Historical offerings