Spring 2020: Linear Algebra II (Math 316)

Overview

Lecture sketches and homework

To help you study for this class, I am posting lecture sketches. These are not a substitute for your personal lecture notes or coming to class (for instance, lots of details and motivation are not included in the sketches). I hope that they are useful to you for revisiting the material and for preparing for exams.

After most classes, homework is assigned and posted below.

• You should aim to complete the problems right after class, and before the next class meets.
  A 15% penalty applies if homework is submitted after the posted due date.
• Homework is submitted online, and you have an unlimited number of attempts. Only the best score is used for your grade.
  Most problems have a random component (which allows you to continue practicing throughout the semester without putting your scores at risk).
• Collect a bonus point for each mathematical typo you find in the lecture notes (that is not yet fixed online), or by reporting mistakes in the homework system. Each bonus point is worth 1% towards a midterm exam.
  The homework system is written by myself in the hope that you find it beneficial. Please help make it as useful as possible by letting me know about any issues!

Videos

Videos complementing the lecture sketches are posted to Panopto and can be accessed in USAonline. Links to the recordings are also included next to the lecture sketches above.

This is my first time making and posting videos, so please share all issues or suggestions!

Sage

As part of this course, we will explore the open-source free computer algebra system Sage to assist with more involved calculations.

If you just want to run a handful quick computations (without saving your work), you can use the text box below.

A convenient way to use Sage more seriously is https://cocalc.com. This free cloud service does not require you to install anything, and you can access your files and computations from any computer as long as you have internet. To do computations, once you are logged in and inside a project, you will need to create a "Sage notebook" as a new file.

Here are some other things to try:

• Sage makes solving least squares problems pleasant. For instance, to solve Example 43 in lecture08.pdf:

  A = matrix([[1,2],[1,5],[1,7],[1,8]]); b = vector([1,2,3,3])
  (A.transpose()*A).solve_right(A.transpose()*b)
  

• Sage can compute QR decompositions. For instance, we can have it do Example 68 in lecture13.pdf for us:

  A = matrix(QQbar, [[0,2,1],[3,1,1],[0,0,1],[0,0,1]])
  A.QR(full=false)
  

  The result is a tuple of the two matrices Q and R. If that is too much at once, A.QR(full=false)[0] will produce Q, and A.QR(full=false)[1] will produce R. (Can you figure out what happens if you omit the full=false? Check out the comment under "Variations" for the QR decomposition in the lecture sketch. On the other hand, the QQbar is telling Sage to compute with algebraic numbers (instead of just rational numbers); if omitted, it would complain that square roots are not available.)
• Sage can also compute singular value decompositions. For instance, Example 150 (see lecture28.pdf) can be done (numerically) as follows:

  A = matrix(RDF, [[2,2],[-1,1]])
  A.SVD()
  

  The result is a tuple of the three matrices U, Σ and V. If that is too much at once, A.SVD()[0] will produce U, A.SVD()[1] will produce Σ, and A.SVD()[2] will produce V. (The RDF is telling Sage to compute with real numbers as floating point numbers with double precision; for other fields such as QQbar, the SVD is not currently implemented.)

An easy way to use Sage more seriously is by creating an account at http://cocalc.com. This free cloud service does not require you to install anything, and you can access your files and computations from any computer as long as you have internet. To do computations, once you are logged in and inside a project, you will need to create a "Sage notebook" as a new file.

Exams and practice material

The following material will help you prepare for the exams.

• Midterm Exam 1:
  Online practice problems (compiled from homework)
  midterm01-practice.pdf, midterm01-practice-solution.pdf
• Midterm Exam 2:
  Online practice problems
  midterm02-practice.pdf, midterm02-practice-solution.pdf
• Final Exam:
  Online practice problems
  final-practice.pdf, final-practice-solution.pdf

Quizzes and solutions

1. quiz01.pdf, quiz01-solution.pdf