Math 342 Spring 2016

Instructor: Robert Lipshitz

Basic course information

Time: MTWF 12:00–12:50 p.m.
Place: 106 Deady Hall.
Textbook: Linear Algebra and Its Applications by David Lay, 5th edition.
Office hours: Wednesday 10:50 - 11:50 and Friday 10:00 - 11:00, 4:00 - 5:00 in Fenton 303. Subject to change.
Final exam: Per the university Final Exam Schedule
Midterm exams: March 28, April 20, May 10. Subject to change if necessary.
There is also WebWork homework, a Canvas webpage for tracking grades and a blog about using computer software with linear algebra.


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



The course will have both written and online 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 online homework assignments due during “dead week”.

You may use any resources you like on the homework, but 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 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.

The WebWork homework site, for online homework, is Written homework assignments are posted below.

A small number of bonus points -- a maximum of 10% of the score on each homework assignment -- will be awarded for following the tutorial on using SageMathCloud alongside the class.


All exams will be given in class, and there will typically not be makeup exams. If you know in advance that you will miss an exam, contact me immediately to make arrangements. If you miss an exam because of an emergency (medical, family, ...) you will be expected to provide documentation of that emergency.

All exams are closed-note, closed-book, and without electronic assistance (including calculators and cell phones). Using any notes or electronic device or communicating with anyone except me during an exam constitutes cheating.

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.


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

Week Date Topic Sections
1 3/28 Introduction to the class. Midterm 0  
  3/29 Review: vector spaces, subspaces, linear transformations 4.1, 4.2
  3/30 Review: span, linear independence, bases, dimension 4.3, 4.5
  4/1 Coordinate systems 4.4
2 4/4 Rank.

This is also the last day to drop the class without a W.
  4/5 Change of basis 4.7
  4/6 Applications: difference equations, Markov chains 4.8, 4.9
  4/8 Review  
3 4/11 Eigenvectors and eigenvalues: definitions, first examples 5.1
  4/12 The characteristic polynomial and how to find eigenvalues and eigenvectors 5.2
  4/13 Diagonalization 5.3
  4/15 Review  
4 4/18 Eigenvectors of linear transformations 5.4
  4/19 Review  
  4/20 Midterm 1  
  4/22 Complex eigenvectors 5.5
5 4/25 Applications of eigenvectors 1: discrete dynamical systems and difference equations 5.6
  4/26 Applications of eigenvectors 2: differential equations 5.7
  4/27 Applications of eigenvectors 3: PageRank 10.1, 10.2
  4/29 Review  
6 5/2 Inner (dot) products, length, orthogonality 6.1
  5/3 Orthogonal sets, Gram-Schmidt process 6.2, 6.4
  5/4 Orthogonal projections 6.3
  5/6 Review  
7 5/9 More review.  
  5/10 Midterm 2  
  5/11 Least-squares optimization 6.5
  5/13 Inner product spaces, Fourier analysis

The last day to withdraw from the class is 5/15.
6.7, 6.8
8 5/16 Diagonalization of symmetric matrices (spectral theorem) 7.1
  5/17 Quadratic forms 7.2
  5/18 Application: constrained optimization 7.3
  5/20 Review  
9 5/23 Singular value decomposition: definition, properties, how to compute it. 7.4
  5/24 Applications of SVD 1: dimensional reduction, eigenfaces (machine learning). 7.5, +
  5/25 Applications of SVD 2: covariance, principal component analysis, ideal supreme court justices. 7.5, +
  5/27 Review  
10 5/30 Memorial day holiday  
  5/31 Introduction to quantum mechanics +
  6/1 Review.  
  6/3 Review. (Last day of classes.)  



Handouts will be posted here, in case you lost the physical copy.


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.