Math 25a: Theoretical linear algebra, Fall 2018


  • (10/19). Homework 6 is posted below.

  • (10/17). Midsemester feedback form here (please complete before class Monday).

  • If you have topics you’d like to hear discussed in section, please submit them here.

  • (9/28). 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.

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 (please let me know if you have a conflict)

  • 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. 

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. 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.

  • Due Sept 14: HW1, solutions (by Raul)

  • Due Sept 21: HW2, solutions

  • Due Sept 28: HW3, solutions

  • No homework due Oct 5. There is a midterm Oct 3. Extra credits 1-3 are due Oct 5.

  • Due Oct 12: HW4, solutions

  • Due Oct 19: HW5

  • Due Oct 26: HW6

  • Due Nov 2:

  • No homework due Nov 9 (there is a midterm Nov 9). Extra credits 4-7 are due Nov 9.

  • Due Nov 16:

  • No homework due Nov 23 (Thanksgiving)

  • Due Nov 30:

  • Due Dec 5:

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.

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

  • Mon. Lecture 12: elementary matrices, inverses

  • Wed. Lecture 13: determinants

  • Fri. Homework 6 due.

Week 9

  • Mon. Lecture 14: more determinants

  • Wed. Lecture 15: eigenvectors and polynomials

  • Fri. Homework 7 due.

Week 10: Axler Ch 4

  • Mon. Lecture 16: polynomials, eigenvector existence

  • Wed. Lecture 17:

  • Fri. Midterm 2. Extra credit due (exact sequences, error-correcting codes, tensor products, alternating forms)

Week 11

  • Mon. Lecture 18: eigenvectors for real operators, satisfied polynomials

  • Wed. Lecture 19: inner products

  • Fri. Homework 8 due.

Week 12

  • Mon. Lecture 20: orthogonality and Gram-Schmidt

  • Wed. Thanksgiving break (no class)

  • Fri.

Week 13

  • Mon. Lecture 21: orthogonal complements and orthogonal projections

  • Wed. Lecture 22: dual spaces and adjoints

  • Fri. Homework 9 due.

Week 14

  • Mon. Lecture 23: spectral theorem

  • Wed. Lecture 24: spectral theorem. Homework 10 due.

  • Fri. Extra credit due. (Google page-rank, universal property)