Math 31AH: Lecture 0

Welcome to Math 31AH, the first quarter of a three-quarter honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Math 31AH focuses on linear algebra, meaning the study of vectors and linear transformations.

Before saying anything else, I want to draw your attention to the following remark which accompanies the registrar’s listing for this course:

The honors calculus courses 31AH-BH-CH are unusually demanding and are intended for students with strong aptitude and deep interest in Mathematics.

– UCSD Registrar’s Office

If you’re enrolled in this class, I want you to be aware of this: Math 31AH is an unusually demanding course. If the prospect of an intellectual challenge appeals to you, you’ve come to the right place; otherwise, this is not the course for you, and you should instead consider Math 18 and the Math 20 course sequence. It is not the case that the Math 31 course sequence is “better” than the Math 18/20 course sequence — but it is different. The Math 31 sequence is more theoretical and rigorous, meaning that there is a strong emphasis on precise definitions and clear logical reasoning, as opposed to simply learning how to use formulas. This does not mean that you are not expected to master formulas in Math 31AH, rather it means that the process of translating theoretical knowledge into the ability to do concrete calculations is largely left to the student — there is a large “make your own examples” component.

Now let us discuss the basic parameters of the course. First of all, we are presently in a highly unusual situation, and will not be able to meet for standard in-person lectures. The course will instead be organized as follows.

This blog will serve as the textbook for the course: the entire course content will be made available here, in the form of lecture posts. These posts will be of a quality unmatched by any textbook, and will only be available to students enrolled in Math 31AH. There will be two posts per week, each of which will contain content corresponding to what would be covered in a rather ambitious in-person lecture. In addition to written text and links to additional material, each of the two weekly lecture posts will be accompanied by a video coda consisting of a discussion of the post together with illuminating examples, worked problems, and sometimes additional content. These videos will be posted to the media gallery section of the Math 31AH Canvas page. Originally I had planned to embed each such video in the corresponding lecture post, but this proved problematic (an example of an embedded video is included at the bottom of this post). Each video coda will be presented in a way which assumes familiarity with the lecture post containing it. The two weekly lecture posts will be made available prior to our designated Monday and Wednesday 15:00-15:50 lecture slots.

That leaves the Friday lecture slot, which will consist of two parts. The first part will occupy the 15:00-15:50 lecture slot, and will be conducted in the style of a flipped classroom in order to cement your understanding of material you have already absorbed. This may take the form of an interactive problem solving session, a recap of course material presented in the lecture posts, or a more free-form discussion of the course material intended to deepen and broaden your understanding. The flipped classroom session will segue into an additional 70 minutes corresponding to “office hours,” which will be driven primarily by student questions (many students find it worthwhile to attend office hours even if they don’t plan to ask questions). The combined flipped classroom and office hours endeavor makes for a two hour live Zoom session every Friday, 15:00-17:00. For those of you not already aware, Zoom is a freely available videoconferencing platform. These Friday Zoom sessions will be recorded and made available on Canvas.

In addition to the course content delivered by Professor Novak in the manner described above, the Math 31AH teaching assistant, Finley McGlade, will be running live discussion sections via Zoom every Thursday, from 11:00-11:50 for students in section A01, and from 12:00-12:50 for students in section A02. Mr McGlade will post further details concerning the discussion sections on Piazza. There will be no discussion sections on October 1.

Piazza is an online platform facilitating online discussion of all aspects of the course, from logistics to course content. If you are enrolled in Math 31AH, you should have already received an email containing Piazza signup instructions. Links to the lecture posts will be made available under the “Resources” section, where you will also find a syllabus link leading back to this post. Both Professor Novak and Mr McGlade will be active on Piazza. We also expect that Piazza will serve as a form for students to discuss the course content with each other, and endeavor to answer each others questions. Please use Piazza as the default mechanism for asking questions about the course, and refrain from using email unless absolutely necessary.

Now that we have discussed how the course content will be disseminated to students, let us discuss the content to be created by students and submitted for evaluation. Students in Math 31AH will generate two types of content: solutions to weekly problem sets, and solutions to exams.

We will aim for a total of nine weekly problem sets in this course. Problem sets will be posted to Piazza on Sundays before 24:00, and the corresponding solutions will be due the following Sunday before 24:00. Your solutions will be both submitted by you and returned to you using Gradescope. This process will be managed by Mr McGlade, and any questions or concerns related to Gradescope should be directed to him. The problem sets are a very important part of the course, and accordingly make up 50% of the total grade. While you may submit handwritten solutions, it is recommended that you typeset your solutions using LaTeX, the professional standard for the preparation of scientific documents. Learning how to prepare scientific documents of professional quality is a valuable skill that will serve you well in your university career, and beyond. In order to help with this, the problem sets will be typeset in LaTeX, and the source files will be posted along with their PDF output. Problem set solutions which have been typeset using LaTeX will receive an automatic 5% bonus.

There will be no problem set due on November 13, because this is the date of the midterm exam. The midterm exam will count for 20% of your course grade. The details of how the midterm exam will be written and submitted are not yet available, but will be soon, and I will keep you updated.

The final exam for the course is set for December 18, with a scheduled time of 15:00-17:59. This date and time slot is set by the university registrar. The final exam will count for 30% of your course grade. The details of how the final exam will be written and submitted are not yet available, but will be soon, and I will keep you updated.

Our first live Zoom session will take place on October 2, 15:00-17:00. I expect you will have many questions, so this meeting will be purely logistical (no math). The schedule below indicates the expected mathematical content of each subsequent lecture. This schedule is subject to change, and may be updated during the quarter.

10/02	Lecture 0	Course logistics.
10/05	Lecture 1	Vector spaces, basis and dimension.
10/07	Lecture 2	Linear transformations, isomorphism, coordinates.
10/09	Lecture 3	Flipped classroom, office hour.
10/12	Lecture 4	Change of basis, Euclidean spaces, Cauchy-Schwarz inequality.
10/14	Lecture 5	Orthogonal bases; Gram-Schmidt algorithm.
10/16	Lecture 6	Flipped classroom, office hour.
10/19	Lecture 7	Orthogonal projection, approximation.
10/21	Lecture 8	Linear forms, bilinear forms.
10/23	Lecture 9	Flipped classroom, office hour.
10/26	Lecture 10	Change of basis, quadratic forms, polarization.
10/28	Lecture 11	Reduction of a quadratic form to a sum of squares.
10/30	Lecture 12	Flipped classroom, office hour.
11/02	Lecture 13	Law of inertia for quadratic forms.
11/04	Lecture 14	Linear transformations and matrices, rank and nullity.
11/06	Lecture 15	Flipped classroom, office hour.
11/09	Lecture 16	Adjoint, symmetric and orthogonal transformations.
11/13	Lecture 17	MIDTERM EXAM.
11/16	Lecture 18	Invariant subspaces, eigenvalues and eigenvectors.
11/18	Lecture 19	Diagonalization of symmetric transformations.
11/20	Lecture 20	Flipped classroom, office hour.
11/23	Lecture 20	Tensor product and wedge product.
11/25	Lecture 21	Wedge product and determinant.
11/30	Lecture 22	Characteristic polynomial, complex numbers.
12/02	Lecture 23	Complex linear algebra.
12/04	Lecture 24	Flipped classroom, office hour.
12/07	Lecture 25	Hermitian and unitary transformations.
12/09	Lecture 26	Normal transformations.
12/11	Lecture 27	Flipped classroom, office hour.