# Math 25a: Theoretical linear algebra, Fall 2018

*Announcements*

Thanks for a great semester!

The website for

**Math 25b 2019**is here.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:

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%)

**Contact Info**

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

*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

*Homework*

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.

Due Sept 28: HW3, solutions, student solutions

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

Due Oct 19: HW5, solutions, student solutions

Due Nov 2: HW7, solutions, student solutions

No homework due Nov 9 (there is a midterm Nov 9). Extra credits 4-6 are due Nov 9.

Due Nov 16: HW8, solutions, student solutions

No homework due Nov 23 (Thanksgiving)

Due Nov 30: HW9, picture, solutions, student solutions

Due Dec 5: HW10 solutions, student solutions

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

EC1: pdf , TeX version (if you use TeX, you’ll need these graphics in the same folder as the TeX file: 1, 2, 3)

EC2: pdf , TeX version

EC3: pdf , TeX version

EC4: pdf, TeX version

EC6: pdf, TeX version

EC8: pdf, link to article (this one reinforces many things discussed in the course and previous extra credits)

*Course Materials*

Main text:

*Linear algebra done right,*by Axler, 3rd editionSupplementary 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!

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)