Fall-2021. Math H110 (class # 22060): Honors Linear Algebra

Instructor: Alexander Givental
Lectures: TuTh 9:30--11 in 385 Physics Bldg
Office hours: TuTh 3:30-5:30 in 701 Evans and on Zoom during the weekend (TBD)
Textbook: Linear Algebra by Alexander Givental
This is an unpublished (and yet unfinished) textbook, which will be made freely available online. One of the previous versions of the text (which will evolve and hopefully finalized during the semester) can be found here.
Quizzes: weekly, each Tuesdays in class (in the first 5 min), so don't be late!
Homework: weekly, due on Thursdays before 9:30 am in person or by email to givental-at-math-dot-berkeley-dot-edy. I strongly recommend that you make some effort to learn bits of LaTeX, and typeset your hw. Think of how much easier it will be for the grader to read your work, and respectively how more adequate the feedback is going to be.

Grading policies:

Here is an ingenious (in my view) scheme of my own invention which I tried successfully in several courses and intend to use it this time. The starting point is: 40% weekly quizzes + 30% weekly hw + 30% final. However: each individual quiz or hw score which is below than or equal to (percentage-wise) your score on the final will be dropped - together with its weight! E.g.: if all your hw scores are above and all quizzes below your score on the final, then your total score is composed of 50% hw and 50% final. Thus, there are many reasons why you want to take quizzes and do hw (as well as many other exercises, not assigned as hw), yet a particular quiz/hw score can only improve your overall performance, but can never hurt your ultimate result compared to the final exam. In our time of many uncertainties caused by the epidemic, this might be particularly useful: if for whatever reason you cannot come to a quiz or are not ready to submit a hw, you should not fret over this -- my policy does not penalize you for skipping it.

Besides, I don't have a preconceived distribution of As,Bs,..., and will be happy to give everyone an A should everyone learn the material well (for which I hope very much, especislly that the subject is interesting and simple). What exactly it means to learn well will become clear after some quizzes.

Thus, the general idea is that you don't compete with each other, but rather strive to learn the material the best you can. Respectively, collaboration and/or use of outside sources are not prohibited (tests excluded). Yet, each instance of this should be explicitly acknowledged in your homework. Failure to acknowledge one's use of somebody else's ideas is commonly known as academic plagiarism. So, let's practice the right ethics.



Chapter 1. Introduction

Chapter 2. Dramatis Personae

Chapter 3. Simple Problems

Chapter 4. Eigenvalues (Sections 1,2)

Chapter 4. Eigenvalues (Sections 3,4)

Hints, answers, index

QZ1: Tu, Aug. 31. This will be a one-question/5-min quiz on Complex numbers at the start of Lecture 2. To prepare, read Section 2 on your own.
HW1: Due Th, Sep. 2: Read Sections 1 and 3. Solve all exercises from Section 1, write down your solutions to the exercises 6,9,22,25,27, and submit them for grading.

QZ2: Tu, Sep. 7. Quadratic curves (Section 3)
HW2: due Th, Sep. 9: Read the rest of Chapter 1. Solve 60, 62, 69, 83, 98.

QZ3: Tu, Sep. 14. Axioms of vector spaces and their consequences.
HW3: due Th, Sep. 16. Read Section 1 (Vector Spaces) of Chapter 2. Solve: 146, 149, 153, 156, 158.

QZ4: Tu, Sep. 21: Matrices of linear maps.
HW4: due Th, Sep. 23. Read Section 2 (Matrices) of Chapter 2. Solve: 169, 172, 174, 188, 193.

QZ5: Tu, Sep. 28: Permutations and their properties.
HW5: due Th, Sep. 30: Read pp. 71-84 (on determiannts). Solve: 195, 212, 220, 221, 223.

QZ6: Tu, Oct. 5. Cramer's Rule.
HW6: due Th, Oct. 7. Read "Determinants" to the end. Solve: 234, 235, 236, 238, 239.

QZ7: Tu, Oct. 12. Read on your own subsection "The Rank Theorem" (pages 101-103)and prepare to a quiz on this material.
HW7: due Th, Oct. 14. Read Section 1 of Ch. 3. Solve: 273, 275, 276, 279, 281.

QZ8: Tu, Oct. 19. Dimension counting.
HW8: due Th, Ot. 21. Read "Gaussian Elimination". Solve: the last system in 288, 289, 290(e), 291(a), and find the number of linear hyperplanes in an n-dimensional vector space over a finite field with q elements.

QZ9: Tu, Oct. 26. Something about classification of pairs of flags.
HW9: due Th, Oct. 28. Read: Section 3 (The Inertia Theorem) from Chapter 3. Solve: 305, 306, 307, 311, 314.

QZ10: Tu, Nov 2: Sylvester's Rule.
HW10: due Th, Nov. 4. Read Section 1 of Chapter 4. Solve: 324 (3rd example), 325, 326 (2nd example), 228, 229.

QZ11: Tu, Nov. 9. Euclidean spaces (read pages 153-154).
HW11: due Th, Nov. 11 by email (you may send photos of your work) . Read: Section 2 of Chapter 4. Solve: 354, 363, 369, 374, 379.

QZ12: Tu, Nov. 16. The geometry of orthogonal transformations.
HW12: due Th, Nov. 18. Read Section 2 of Chapter 4. Solve: 390, 400(b), 401, 403, 408.

QZ13: Tu, Nov. 23. Let's skip the quiz as well as HW this week but: To prepare for Tuesday's lecture, read before Tuesday pages 169-173 of the newly posted portion of the text.

QZ14: Tu, Nov. 30. Root spaces.
HW14: Th, Dec. 2: Read Chapter 4, part 2 to the end. Solve: 421, 423, 427(d), 432, 434

HW Solutions