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.
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 (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.
No homework due Oct 5. There is a midterm Oct 3. Extra credits 1-3 are due Oct 5.
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.
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.
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.
Mon. Lecture 12: elementary matrices, inverses
Wed. Lecture 13: determinants
Fri. Homework 6 due.
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)
Mon. Lecture 18: eigenvectors for real operators, satisfied polynomials
Wed. Lecture 19: inner products
Fri. Homework 8 due.
Mon. Lecture 20: orthogonality and Gram-Schmidt
Wed. Thanksgiving break (no class)
Mon. Lecture 21: orthogonal complements and orthogonal projections
Wed. Lecture 22: dual spaces and adjoints
Fri. Homework 9 due.
Mon. Lecture 23: spectral theorem
Wed. Lecture 24: spectral theorem. Homework 10 due.
Fri. Extra credit due. (Google page-rank, universal property)