Math 25a: Theoretical linear algebra, Fall 2018
Thanks for a great semester!
The website for Math 25b 2019 is here. We are going to use some subset of the books below.
Pugh, Real mathematical analysis
Munkres, Analysis on manifolds
Spivak, Calculus on manifolds
Hubbard-Hubbard, Vector calculus, linear algebra, and differential forms
These are ordered by how much I anticipate we’ll use them. However, things might change, so I wouldn’t necessarily recommend buying any of them yet. See the course website for more information.
In this course, we'll pursue a rigorous treatment of linear algebra. Topics include set theory, vector spaces and bases, linear maps, determinants, eigenvectors, inner products, and spectral theory. The plan is to work through Axler's Linear algebra done right, although we will use some other resources as well (see Course Materials below). This course is part one of a two-semester journey -- the second semester will be real analysis and multivariable calculus.
The goals of the course include:
Learning how to read and write proofs, and how to critique an argument. Learning to think carefully, logically, and rigorously. Learning to communicate math clearly and effectively.
Learn both theoretical and computational aspects of linear algebra. More emphasis will be on the former, although both are important!
Learn to use LaTeX. See the section on Homework below.
Create a fun environment/community for learning and discussing math.
This course has no formal prerequisites -- we'll start from the basics. Some exposure to linear algebra or proofs is helpful but definitely not required. You don't need to be a math olympiad, nor do I think this will be particularly helpful. More important is a strong desire to learn mathematics. This course is fast-paced and time intensive. Homework sets require a significant amount of work, and for practical purposes, they can't be done completely by yourself -- you'll need/want to collaborate. A typical weekly assignment may take 10-15 hours to complete, perhaps more early in the semester. That said, the course should also be a lot of fun, and if you focus on mastering the material, you will learn a lot.
Grading: Weekly homework (30%), two midterms (40%), final exam (30%)
Bena Tshishiku (Lecturer) bena at math.harvard.edu
Raul Chavez-Sarmiento (GCA) rchavez at math.harvard.edu
Joseph Feffer (CA) jrfeffer at college.harvard.edu
Davis Lazowski (CA) dlazowski at college.harvard.edu
Beckham Myers (CA) bmyers at college.harvard.edu
Laura Zharmukhametova (CA) lzharmukhametova at college.harvard.edu
Class: MW 9-10:15am
Section: Tuesdays 4:30-5:30 in SC 530. Submit questions here.
Math Night (collaborative homework session): Monday 8-10pm, Leverett House Dining Hall (more info)
Monday: (Davis) 8-10p in Leverett dining hall (math night)
Tuesday: (Bena) 3-4p in SC 111
Wednesday: (Bena) 10:30-11:30a in SC 530, (Laura) 8-10p in Lowell dining hall
Thursday: (Raul) 3-4 in SC 428e, (Beckham) 8:30-10p in Lowell dining hall
Friday: (Joey) 9-10:15a in SC 411
Or by appointment (email me).
Important dates (see also here)
Registration deadline: Wed, Sept 12
Drop-without-fee deadline: Mon, Sept 24
Drop deadline: Oct 9 (last possible day to switch between 23/25/55/etc)
Midterm 1: Wed, Oct 3
Midterm 2: Fri, Nov 9, 9-10:15am
FInal exam: Dec 12, 2-5p
There will be 10 assignments posted below as the semester progresses. Homework will be due each Friday at 5pm in the CA's mailboxes on the 2nd floor of the Science Center.
Late homework policy: For your homework grade, I will drop the score from your lowest assignment. View this as a one-time "get out of jail free card" in the event that you oversleep, have a midterm, etc. As a rule late homework will not be accepted. If you have a medical circumstance that prevents you from turning in homework, you will need a doctor's note to receive an extension. Furthermore, a note that merely says “Student visited HUHS today” is not very helpful; it’s better to have a note that says something like “Student may be unable to complete assignments for the next N days.” This will help me decide what length of extension is appropriate.
Collaboration: You are encouraged to work together on the assignments (c.f. Math Night). In your solutions, you should acknowledge the students with whom you worked, and you should write your solutions on your own. For any solution you submit, you should understand it well enough that you can explain it to someone else and answer questions about it. If you find yourself writing down things that you can’t explain, you should go back and think more about the problem. Otherwise you’re doing yourself a disservice, and you might suffer more when it comes time to take the midterms/exam (note that HW is only 30% of the final grade).
LaTeX: It is required that you type your solutions in LaTeX. You can either download LaTex or use sharelatex which allows you download, edit, and compile LaTeX files online. See here for a guide to notation in LaTeX. Also Detexify is a useful tool for finding the commands for various symbols. The source code for the assignments should be a helpful guide. The most basic thing to know/remember is that math always goes in between dollar signs.
Extra credit: Throughout the semester there will be several extra credit opportunities, which will allow you to explore topics beyond the lectures/textbook. These assignments are meant to be fun, and the extra points might help you to not stress too much about your grade. The due dates will be on the Friday after each midterm and also the last week of class. Please note that the extra credit is “extra”, i.e. optional, and if your HW/exam scores are good, you won’t need the extra credit to get a good grade in the course.
No homework due Oct 5. There is a midterm Oct 3. Extra credits 1-3 are due Oct 5.
No homework due Nov 9 (there is a midterm Nov 9). Extra credits 4-6 are due Nov 9.
No homework due Nov 23 (Thanksgiving)
Extra credit assignments. I’ve included the TeX version, but you do not have to write your solutions in TeX.
Main text: Linear algebra done right, by Axler, 3rd edition
Supplementary videos made by Axler to accompany the text
Problem solving strategies by George Melvin
Supplementary texts. When learning a topic, it's always a good idea to have multiple sources!
Midterm 1 materials.
Axler’s anti-determinant manifesto: Down with Determinants!
Midterm 2 materials.
The Mathematica notebook on polynomial approximation is here.
Final exam materials.
The final exam is cumulative, but will focus on material since the last midterm (inner products, adjoints, spectral theorem, positive and orthogonal operators; also eigenvalues, characteristic polynomial, determinant). I’m not going to ask any set theory questions other than the main definitions/theorems.
You should know the statements of the big definitions and theorems from the course. I’m not going to ask you obscure things — if it appeared more than twice during the course, then it’s fair game!
You should know how to prove some of the major theorems from the course. One of the questions on the exam will ask you to prove a theorem we proved in class.
It might be helpful to look back over old homework assignments and make sure you understand them. I like giving problems that have some resemblance to something you already did.
An easy way to remember the definition of an inner product.
If you have topics you’d like to hear discussed in section, please submit them here.
Homework comments/tips from the CAs can be found here.
Topic schedule (tentative)
This schedule will be updated as we go along. In general we are following the trajectory of Axler, but at times we will do things differently. We won't be able to cover everything in the corresponding sections of Axler in class. It's worth noting that the homework assignments also contain pointers to sections of the text.
Week 1: Simmons Ch 1, sections 1-5
Mon. Labor day (no class)
Wed. Lecture 1: sets, functions, cardinality
Week 2: Simmons Ch 1, sections 5-7
Mon. Lecture 2: countability, equivalence relations
Wed. Lecture 3: equivalence relations, more cardinality
Fri. Homework 1 due.
Week 3: Axler Ch 1
Mon. Lecture 4: fields, vector spaces (definition, examples)
Wed. Lecture 5: subspaces, direct sums, spanning sets
Fri. Homework 2 due.
Week 4 : Axler Ch 2
Mon. Lecture 6: bases and dimension
Wed. Lecture 7: linear independence theorem
Fri. Homework 3 due.
Week 5: Axler Ch 3, sections 3A-3B
Mon. Lecture 8: linear maps, kernel/image
Wed. Midterm 1.
Fri. Extra credit due (Hilbert hotel, prime numbers, quotient spaces)
Week 6: Axler Ch 3, section 3C
Mon. Columbus day (no class).
Wed. Lecture 9: matrices, rank-nullity
Fri. Homework 4 due.
Week 7: Axler Ch 3, section 3D; Treil Ch2
Mon. Lecture 10: matrix multiplication, invertibility, linear systems
Wed. Lecture 11: linear systems, row operations
Fri. Homework 5 due.
Week 8: Treil Chapters 2 and 3.
Mon. Lecture 12: elementary matrices, inverses
Wed. Lecture 13: determinants
Fri. Homework 6 due.
Week 9: Treil Chapter 3, 4; Axler Ch 5
Mon. Lecture 14: more determinants
Wed. Lecture 15: eigenvectors and polynomials
Fri. Homework 7 due.
Week 10: Treil Ch 4 (also to a lesser extent Axler Ch4)
Mon. Lecture 16: polynomials, eigenvector existence, diagonalizability
Wed. Lecture 17: more eigenvectors, diagonalizability
Fri. Midterm 2. Extra credit due (exact sequences, error-correcting codes, tensor products, alternating forms)
Week 11: Treil Ch 5 and Axler Ch 6
Mon. Lecture 18: spectral theorem and inner products
Wed. Lecture 19: inner products, orthogonality
Fri. Homework 8 due.
Week 12: Treil Ch 5 and Axler Ch 6
Mon. Lecture 20: orthogonal complements, projections, and polynomial approximation
Wed. Thanksgiving break (no class)
Week 13: Treil Ch 6 and Axler Ch 7
Mon. Lecture 21: dual spaces and adjoints
Wed. Lecture 22: spectral theorem
Fri. Homework 9 due.
Week 14: Treil Ch 6 and Axler Ch 7
Mon. Lecture 23: spectral theorem
Wed. Lecture 24: spectral theorem. Homework 10 due.
Fri. Extra credit due. (Google page-rank, universal property)