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.

Course 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: 

  1. 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.

  2. Learn both theoretical and computational aspects of linear algebra. More emphasis will be on the former, although both are important!

  3. Learn to use LaTeX. See the section on Homework below.

  4. 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%)

Contact Info

  • Bena Tshishiku (Lecturer) bena at

  • Raul Chavez-Sarmiento (GCA) rchavez at

  • Joseph Feffer (CA) jrfeffer at

  • Davis Lazowski (CA) dlazowski at

  • Beckham Myers (CA) bmyers at

  • Laura Zharmukhametova (CA) lzharmukhametova at

Course Events

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)

Office Hours

  • 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.

Extra credit assignments. I’ve included the TeX version, but you do not have to write your solutions in TeX.

Course Materials

  • 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

  • Proof-writing guide by Eugenia Cheng

  • Supplementary texts. When learning a topic, it's always a good idea to have multiple sources!

    • At the beginning of the course we will cover some set theory that's discussed in Introduction to topology and modern analysis by Simmons. I suggest you avoid buying it and use the digital copy found here.

    • Throughout the course we will use Linear algebra done wrong by Treil. Find a digital copy here.

  • 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.

    • Practice 1, 2, 3 and solutions 1, 2, 3. Comments about the exam:

    • 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.

  • Math extracurricular:

    • Math Table, Tuesdays at 6p with free food.

    • Open Neighborhood Seminar, some Mondays 4:30p.

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

  • Fri.

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)

  • Fri.

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)