Math 25a: Honors Linear Algebra and Real Analysis I, Fall 2017


  • For the 25a page for 2018 go here
  • (12/15): Final exams will be available at the beginning of the next semester.
  • (12/13): If you are planning to take 25b, we will be using Spivak's Calculus on manifolds. More information will be available soon. We will likely also use the following as supplementary texts:
    • Munkres, Analysis on manifolds
    • Hubbard and Hubbard, Vector calculus, linear algebra, and differential forms
  • (12/12): The exam is tomorrow in Emerson 210. It will start promptly at 2pm. 
  • (12/3): The exam review is posted here. Practice exams here and here and here. They should give you a good idea for what the exam will look like. Practice exam solutions are posted here, here, and here

Course Information

A rigorous treatment of linear algebra. Topics include: construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors; determinants and inner products. The plan is to work through Axler's Linear algebra done right after a short introduction using Chapter 1 of Simmons' Introduction to topology and modern analysis. This course is part one of a two-part course -- the plan for the second semester is to work through Spivak's Calculus on manifolds

Grading: Weekly homeworks (30%), two midterms (40%), final exam (30%)

Contact Info

  • Bena Tshishiku (Lecturer) bena at
  • Yifei Zhao (Teaching fellow) yifei at
  • Ellen Li (CA) eyli at
  • Charles O'Mara (CA) comara at
  • Michele Tienni (CA) micheletienni at
  • Natalia M. Pacheco-Tallaj (CA) pachecotallaj at

Course Events

Class: MWF 10-11am in SC 507

Section: Thursday 7-8pm in SC 112. Please submit questions here

Math Night (collaborative homework session): Monday 8-10pm, Leverett House Dining Hall (more info)

Office Hours. 

  • Bena: Monday and Friday 3-4 in SC 232, or by appointment in SC 525
  • Yifei: Thursday 4-5 in SC 321g
  • Ellen & Natalia: Monday 8-10p, Leverett dining hall (Math night)
  • Michele & Charlie: Tuesday 8:30-10:30p, Lowell dining hall 

Important Dates

  • Drop deadline: Monday, Sept 18
  • Deadline to switch between 21/23/25/55: Monday, Oct 2
  • Midterm 1: Wednesday, Sept 27
  • Midterm 2: Wednesday, Nov 1
  • Final exam: December 13, 2-5 pm in Emerson 210


There will be 10 assignments posted below as the semester progresses. Homework will be due each Wednesday before 10am in the CA's mailboxes on the 2nd floor of the Science Center. 

Late homework policy: As a rule late homework will not be accepted. If you have a medical issue 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. 

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. 

Exploration problems: The exploration problem is intended for those student who want an additional challenge beyond the homework. They give you an opportunity to travel deeper into abstraction. For this journey, it's best if you're accompanied by an "adult". For this reason, you are highly encouraged to meet with Yifei or me about the exploration problems. This can be during office hours or by appointment. There will be no deadline for the  problems. Turn them in when you're ready. Feel free to go back and revisit the first or second exploration and please come talk to me or Yifei about it! 

  • Due Sept 6: HW1
  • Due Sept 13: HW2, exploration 1 (designed by Yifei -- for extra credit)
  • Due Sept 20: HW3, exploration 2 (due Oct 4)
  • No homework due Sept 27. Instead please work on the practice midterm problems. It won't be turned in, but you are strongly encouraged to do all of them!
  • Due Oct 4: HW4
  • Due Oct 11: HW5
  • Due Oct 18: HW6, exploration 3,
  • Due Oct 25: HW7 you will need to add this and this file to the folder that you put the TeX file in.
  • No homework due Nov 1 (there will be a midterm). Here is a short study guide. Here are practice midterms 1, 2, 3
  • Due Nov 8: HW8
  • Due Nov 15: HW9, exploration 4
  • No homework due Nov 22 (Thanksgiving)
  • Due Nov 29: HW10

Course Materials

  • Lecture notes taken by Michele. 
  • The CAs have collected some comments about common homework mistakes here.  
  • Read more about distance geometry here
  • Read more about the matrix-tree theorem here
  • If you're curious why determinants appear at the very end of Axler, read his article "Down with determinants" here
  • YBC 7289 is an ancient tablet that illustrates the Babylonians' ability to take square roots. Apparently, it's at Yale, so maybe check it out while you're at the Harvard-Yale game(!?). 
  • Information about summer research programs at Harvard and elsewhere. 
  • Solutions to Midterm 2
  • Math advice from Terry Tao's blog 
  • Polynomial approximation mathematica sheet
  • Google page-rank algorithm article
  • Google doc with comments on the homework, written by the CAs. 
  • Problem solving strategy (written by George Melvin)
  • Solutions to Midterm 1
  • Main textbook: Linear algebra done right (3rd edition) by Axler. This is the main text of the course. It's available in the book store.
  • Additional text: Introduction to topology and modern analysis by Simmons. The first two weeks of the course will be centered around the material in Chapter 1.  I suggest you avoid buying it and use the digital copy found here
  • Supplementary videos (by Axler) accompanying the text

Topic schedule

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 7 (Oct 9-13): Ch 4, 5A
    • Mon: Columbus day
    • Wed: Invertibility, eigenvectors and eigenvalues, the root theorem
    • Fri: Division algorithm for polynomials, application to roots
  • Week 8: 5A-C
    • Mon: fundamental theorem of algebra, eigenvectors for operators on complex vector spaces
    • Wed: eigenvectors for operators on real vector spaces
    • Fri: eigenvectors for operators on real vector spaces / Google's page-rank algorithm
  • Week 9: 5C, 6A-B
    • Mon: satisfied polynomials, eigenvalues, inner products
    • Wed: properties of inner products: Pythagorean theorem, Cauchy-Schwarz, triangle inequality
    • Fri: orthonormal bases, Gram-Schmidt algorithm, orthogonal complements
  • Week 10: 6C, 7A
    • Mon: approximating functions by polynomials: orthogonal projections and Gram-Schmidt  
    • Wed: Midterm
    • Fri: adjoints and the spectral theorem 
  • Week 11: 7A-C
    • Mon: More on adjoints, proof of spectral theorem (part 1)
    • Wed: proof of spectral theorem (part 2), positive operators, isometries
    • Fri: square roots in L(V), positive operators
  • Week 12: 7C-D
    • Mon: inner products and positive matrices, eigen-decomposition 
    • Wed: polar decomposition 
    • Fri: singular value decomposition
  • Week 13 (Thanksgiving): 
    • Mon: determinants
  • Week 14: 
    • Mon: determinants, part 2
    • Wed: determinants and spanning trees
    • Fri: distance geometry (looking back and looking ahead)