Math 342 Spring 2020

Instructor: Robert Lipshitz. CRN: 33623.

Basic course information

Time: MTWF 2:00–2:50 p.m.
Place: Online, initially via Zoom.
Textbook: Linear Algebra and Its Applications by David Lay, 5th edition.
Office hours: Mondays 12-12:50, Wednesdays 3-3:50, Fridays 12-12:50 online, via Zoom. Subject to change.
Final exam: Per the university Final Exam Schedule.
Midterm exams: April 1 and 20, May 8. Subject to change if necessary.
There is also WebWork homework, a Canvas webpage, and a blog about using computer software with linear algebra.

Remote Learning Format

The plan for how I will run this remote course is below. All of it is subject to change, as I find out what does and does not seem to work well. I am optimistic that this online format will be as effective for learning the material as a traditional class, though it will take a little more effort for all us to stay focused and engaged.


Lectures will be given over Zoom. Mainly, I will write in a shared OneNote "class notebook", and share my screen while doing so. You can (and are encouraged to) take your own notes during class, but you will also have access to a copy of what I write, inside OneNote. The backup plan is for me to instead write on paper and share that with you via a document camera, and later upload scans of what I wrote.

An invitation to the Zoom lectures ("meetings") will be posted to Canvas. It will probably also be sent by e-mail to the list of enrolled students. You will also be automatically added to the OneNote class notebook.

Assuming everything is working properly, the Zoom lectures will be recorded and posted to Canvas (via Panopto). I do not guarantee that I will be able to post recordings of every Zoom lecture -- don't skip lecture because you are counting on being able to watch the recording.

The Zoom lectures will also be interactive: you will be encouraged to ask questions (via voice or chat), and I plan to break the class up into small discussion groups to work on problems and discuss concepts during class. Be prepared to interact with me and other students. In particular, you must have a working microphone.

To prepare, before the quarter starts:

  1. Make sure you have Zoom working on your computer or tablet. (A phone is also an option in the worst case, but it may be hard to see what I write.) Make sure you know how to mute and un-mute your microphone.
  2. Install OneNote on your computer or tablet, or make sure you can log in to OneNote on the web. Log in to OneNote with your DuckID.
  3. If possible, find a place that you are comfortable joining the lecture with your video turned on: you will probably find it easier to pay attention if your video is on, and the visual feedback I receive that way helps me pace the lectures.

Part of the first few lectures will be spent getting used to learning through Zoom.

Fridays will typically be used for review and homework help, as well as catching up if (when) we get behind on material.

Office Hours

Office hours will also be via Zoom, in a similar format to the lectures. They will not be recorded, but I may make part or all of what is written during them available in OneNote.


The class will have both written homework and WebWorks homework. Written homework should be uploaded to Canvas by the deadline. You can solve the homework by:

  1. Solving the problems on paper and then scanning your solutions. You can use a scanning app like Adobe Scan, Office Lens, Scanner Pro, Scanbot, or others, but your scans should be easy to read and contained in a single file. If you do not have a tablet, this is probably the easiest option.
  2. Writing solutions using a tablet (iPad, Surface, etc.) and exporting and uploading the result. If you have a tablet, this is probably the easiest option.
  3. Typing your solutions using LaTeX, Word, or Pages. You need to show your work, so typing your solutions is a substantial amount of effort. I would do this only if you're trying to learn to type mathematics in LaTeX.

Graded homework will be returned electronically, somehow -- maybe using Canvas, but perhaps using Dropbox or OneNote.

WebWorks homework is as usual.


Tentatively, I plan to give two in-class midterms, an in-class placement / review exam ("Midterm 0", in the first week), and a take-home final exam. All exams will be open-note and open-book. They are to be completed individually -- you may not consult with anyone else (except me) about linear algebra while taking the exams.

Midterm 0 will be via WebWorks, for 25 minutes on Wednesday of week 1. Make sure you know how to log in to WebWorks and answer questions using it before then.

The other midterm exams will either be via WebWorks or by uploading your solutions to Canvas, depending on how Midterm 0 goes.

I plan to give you roughly 48 hours for the final exam.

Required and Recommended Software and Hardware

To summarize, you need:

  • A computer or tablet from which you can access Zoom, including a microphone you can use to talk on Zoom.
  • A way to create electronic copies of your homework solutions -- either by scanning paper or writing your solutions electronically.

It would be helpful, but is not absolutely, to have:

  • A webcam and a place to participate in lectures and office hours where you are comfortable being on video. (Note: if you have a sufficiently powerful computer, Zoom can obscure your background.)
  • A tablet and stylus, or other way of writing in OneNote or Zoom's whiteboard (or using Zoom's annotation tools).



Math 341 (first quarter linear algebra).

Description and goals

This course continues the study of linear algebra where Math 341 left off. The course has two main topics, eigenvalues / eigenvectors and inner products, both of which are key tools in science, engineering, and more advanced mathematics. Applications, which we will touch on in the class, range from the formulation of quantum mechanics to the core of Google's webpage ranking algorithm.

Specific “learning outcomes” include:

  • Developing an understanding of the notion of abstract vector spaces and linear transformations, and how to represent vectors and linear transformations in terms of bases.
  • To understand how the matrix for a linear transformation when one changes bases.
  • To understand the notion of eigenvalues and eigenvectors and use them to diagonalize matrices (when possible) and understand qualitative and quantiative properties of matrices, linear transformations, differential equations, and difference equations.
  • To understand the spectral theorem for symmetric matrices and its application to quadratic forms, and applications of quadratic forms, in turn, to optimization problems.
  • To be able to find and exploit orthonormal bases.

A course is more than its “learning outcomes”: the goal is understanding, not the ability to perform specific manipulations.



Written homework 20%
Online homework 10%
Class participation 5%
Midterm 0 5%
Midterm 1 15%
Midterm 2 15%
Final 30%



The course will have both written and WebWorks homework. Written homework is due at the beginning of class on Mondays, except as noted. Online homework, via WebWorks, is due before class on Mondays, except as noted. (Due dates may change.) There will be written and WebWorks homework assignments due during “dead week”.

You may use any resources you like on the homework, but for written homework all resources except the textbook must be cited on your assignment. This includes help from your classmates, friends, or Google. Failure to cite sources constitutes plagiarism, a serious form of academic dishonesty, and will be punished.

You may work together on homework assignments, but you must write up the final version of your answers by yourself. Working on the final write-ups together constitutes cheating.

Late homeworks will typically not be accepted, but the lowest written homework score and lowest online homework score will be dropped. Due to limited resources, only part of the homework will be graded carefully.

This is the link to the WebWork homework site, for online homework. Written homework assignments are posted below.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Written homework

Again, all written homework is due at the beginning of class on the due date. Remember that there is also online homework.


This schedule is tentative, and may change during the quarter.

Week Date Topic Sections
1 3/30 Introduction to the class. Review: vector spaces, subspaces, linear transformations 4.1, 4.2
  3/31 Review: span, linear independence, bases, dimension. 4.3, 4.5
  4/1 Midterm 0. Continue review.  
  4/3 Coordinate systems. 4.4
2 4/6 Rank. Homework 1 due.

This is also the last day to drop the class without a W.
  4/7 Change of basis. 4.7
  4/8 Applications: difference equations, Markov chains. 4.8, 4.9
  4/10 Review  
3 4/13 Eigenvectors and eigenvalues: definitions, first examples. Homework 2 due. 5.1
  4/14 The characteristic polynomial and how to find eigenvalues and eigenvectors. 5.2
  4/15 Diagonalization. 5.3
  4/17 Review.  
4 4/20 Midterm 1. Homework 3 due.  
  4/21 Eigenvectors of linear transformations. 5.4
  4/22 Review of complex numbers. Appendix B
  4/24 Complex eigenvectors. 5.5
5 4/27 Applications of eigenvectors 1: discrete dynamical systems and difference equations. Homework 4 due. 5.6
  4/28 Applications of eigenvectors 2: PageRank. 10.1, 10.2
  4/29 Inner (dot) products, length, orthogonality. 6.1
  5/1 Review.  
6 5/4 Orthogonal sets, Gram-Schmidt process. Written Homework 5 due. 6.2, 6.4
  5/5 Orthogonal projections. 6.3
  5/6 Review. Online Homework 5 due.  
  5/8 Midterm 2.  
7 5/11 Least-squares optimization. Written Homework 6 due. 6.5
  5/12 Inner product spaces, Fourier analysis. 6.7, 6.8
  5/13 Applications of eigenvectors 3: differential equations. 5.7
  5/15 Review.

The last day to withdraw from the class is 5/20.
8 5/18 Diagonalization of symmetric matrices (spectral theorem). Homework 7 due. 7.1
  5/19 Quadratic forms. 7.2
  5/20 Application: constrained optimization. 7.3
  5/22 Review.  
9 5/25 Memorial day holiday  
  5/26 Singular value decomposition: definition, properties, how to compute it. Homework 8 due. 7.4
  5/27 More on SVD. 7.5
  5/29 Review.  
10 6/1 Applications of SVD 1: dimensional reduction, eigenfaces (machine learning). Homework 9 due. +
  6/2 Applications of SVD 2: covariance, principal component analysis, ideal supreme court justices. +
  6/3 Review.  
  6/5 Review. (Last day of classes.)  




Reading mathematics. You are expected to read the sections in the textbook before coming to class. It's usually only a few pages, so read it carefully. Note down the questions you have; I would expect you to have at least one per page. Read the section again after class. See which questions you now understand. Think about the remaining questions off and on for a day. See which you now understand. Ask someone (e.g., me) about the questions you still have left.

Getting help. If you're having trouble, get help immediately. Everyone who works seriously on mathematics struggles, but if you don't get help promptly you will soon be completely lost. The first places to look for help are my office hours. The Teaching and Learning Center also facilitates individual and small-group tutoring.

Teaching to learn. The best way to learn mathematics is to explain it to someone. You'll find that, particularly in office hours, I'll try to get you to explain the ideas. You should also try explaining the material to each other. The person doing the explaining will generally learn more than the explainee. Another thing to try is writing explanations to yourself, in plain English or as close as you can manage, of what's going on in the course. File them somewhere, and then look back at them a few days later, to see if your understanding has changed.