arminstraub.com

# Spring 2019: Linear Algebra II (Math 316)

## Overview

 Instructor Dr. Armin Straub MSPB 313 straub@southalabama.edu (251) 460-7262 (please use e-mail whenever possible) Office hours MWF, 9-10am, 11am-noon, or by appointment Lecture MWF, 8:00-8:55am, in MSPB 410 Midterm exams The tentative dates for our two midterm exams are: Friday, February 15 Wednesday, March 27 Final exam Monday, April 29 — 8:00am-10:00am Text Linear Algebra with Applications, by Jeffrey Holt (W. H. Freeman, 2013) Online grades USAonline Syllabus syllabus.pdf

## 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!
Date Sketch Homework
01/07 lecture01.pdf Homework Set 1: Problems 1-5 (due 1/18)
01/09 lecture02.pdf Homework Set 1: Problem 6 (due 1/18)
01/11 lecture03.pdf Homework Set 1: Problems 7-8 (due 1/18)
01/14 lecture04.pdf Homework Set 2: Problems 1-3 (due 1/27)
01/16 lecture05.pdf Homework Set 2: Problems 4-5 (due 1/27)
01/18 lecture06.pdf Homework Set 2: Problems 6-9 (due 1/27)
01/23 lecture07.pdf Homework Set 3: Problems 1-3 (due 2/3)
01/25 lecture08.pdf Homework Set 3: Problems 4-6 (due 2/3)
01/28 lecture09.pdf Homework Set 4: Problems 1-2 (due 2/8)
01/30 lecture10.pdf Homework Set 4: Problems 3-4 (due 2/8)
02/01 lecture11.pdf Homework Set 4: Problems 5-6 (due 2/8)
02/04 lecture12.pdf Homework Set 5: Problems 1-2 (due 2/15)
02/06 lecture13.pdf additional homework next class
02/08 lecture14.pdf review quiz (solutions below); do Example 74
02/11 lecture15.pdf Homework Set 5: Problems 3-4 (due 2/15)
02/13 review get ready for the midterm exam on Friday:
midterm01-practice.pdf (solutions below)
02/18 lecture16.pdf Homework Set 6: Problems 1-2 (due 3/6)
02/20 lecture17.pdf Homework Set 6: Problem 3 (due 3/6)
02/22 lecture18.pdf Homework Set 6: Problem 4 (due 3/6)
02/25 lecture19.pdf Homework Set 6: Problem 5 (due 3/6)
02/27 lecture20.pdf Homework Set 6: Problems 6-8 (due 3/6)
03/01 lecture21.pdf Homework Set 7: Problems 1-3 (due 3/15)
03/04 lecture22.pdf Homework Set 7: Problem 4 (due 3/15)
03/06 lecture23.pdf Homework Set 7: Problems 5-7 (due 3/15)
03/08 lecture24.pdf Homework Set 7: Problem 8 (due 3/15)
03/11 lecture25.pdf Homework Set 8: Problems 1-2 (due 4/8)
03/13 lecture26.pdf Homework Set 8: Problem 3 (due 4/8)
03/15 (bonus) quiz Happy Spring break!
03/25 review get ready for the midterm exam on Wednesday:
midterm02-practice.pdf (solutions below)
03/29 lecture27.pdf Homework Set 8: Problem 4 (due 4/8)
review exam
04/01 lecture28.pdf Homework Set 8: Problems 5-7 (due 4/8)
04/03 lecture29.pdf finish homework; new homework next class
04/05 lecture30.pdf Homework Set 9: Problem 1 (due 4/22)
04/08 lecture31.pdf Homework Set 9: Problems 2-3 (due 4/22)
04/10 lecture32.pdf new homework next class
04/12 lecture33.pdf Homework Set 9: Problems 4-6 (due 4/22)
04/15 lecture34.pdf Homework Set 9: Problem 7 (due 4/22)
04/17 lecture35.pdf Homework Set 10: Problem 1 (due 4/26)
04/19 lecture36.pdf Homework Set 10: Problems 2-3 (due 4/26)
04/22 lecture37.pdf Compute the 3rd and 4th Legendre polynomial (see Example 168)
04/24 lecture38.pdf start preparing for the final exam
04/26 review get ready for the final exam!
final-practice.pdf (solutions below)

## 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.
(The pre-entered code shows how to solve Example 44 in lecture08.pdf.)

An easy way to use Sage more seriously is https://cocalc.com. This free cloud service does not require you to make an account or to install anything: after choosing Use CoCalc Anonymously, select View Your CoCalc Projects... (at the bottom of the page), create a project (choose any name for it), followed by New and Sage worksheet and start computing. (To save your work for later, you can create a free account.)

Here are some other things to try:

• Sage can compute QR decompositions. For instance, we can have it do Example 64 (see lecture12.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 lecture12.pdf. 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 144 (see lecture30.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.