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


  • (11/10): There is a reference about the Babylonian method under course materials. 
  • Google form for the weekly section here. Please submit questions! 
  • (11/8): Homework 9 correction/suggestions:
    • The vector space R^n has a standard inner product (i.e. the dot product, i.e. the inner product where the standard basis is orthonormal). When a problem discusses R^n as an inner product space (like problem 1), the inner product is assumed to be the standard one. 
    • Problem 6: there is a mistake in Axler's statement. The norm should be defined as
      ||(x,y)||=( |x|^p + |y|^p )^{1/p}. As a hint, you can solve this problem using one of the other problems on the assignment. (Hm, which one??)
    • Problem 11: You will need the identity (ST)*=T*S*. We didn't prove it in class, so you should prove it!
  • (10/30): The mathematica work we did today is available under Course Materials. 
  • (10/14): The CAs have collected some comments about common homework mistakes 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, 2pm


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. New: There will be no deadline for the exploration 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! 

Course Materials

  • 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: more determinants
  • Week 14: 
    • Mon: determinants and characteristic polynomials
    • Wed: determinants and volume
    • Fri: singular value decomposition application