arminstraub.com

# Spring 2018: 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 MW, 9:00am-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 16 Wednesday, March 21 Final exam Monday, April 30 — 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).
• You are the first to test, and hopefully benefit from, this system, which I have written myself over the break. Please report any issues to make it as useful as possible!
• 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.
Date Sketch Homework
01/08 lecture01.pdf Homework Set 1: Problems 1-5 (due 1/17)
01/10 lecture02.pdf Homework Set 1: Problems 6-8 (due 1/17)
See lecture notes for the full details on the diagonalization of the matrix from class.
01/12 lecture03.pdf complete the previous homework (due on Wednesday!)
01/19 lecture04.pdf Homework Set 2: Problems 1-3 (due 1/29)
01/22 lecture05.pdf Homework Set 2: Problems 4-5 (due 1/29)
01/24 lecture06.pdf Homework Set 3: Problems 1-4 (due 2/7)
01/26 lecture07.pdf Homework Set 3: Problems 5-6 (due 2/7)
01/29 lecture08.pdf Homework Set 3: Problems 7-9 (due 2/7)
01/31 lecture09.pdf Homework Set 4: Problems 1-2 (due 2/16)
02/02 lecture10.pdf Homework Set 4: Problem 3 (due 2/16)
02/05 lecture11.pdf Homework Set 4: Problem 4 (due 2/16)
02/07 lecture12.pdf Homework Set 4: Problems 5-7 (due 2/16)
02/09 lecture13.pdf start preparing for the midterm exam next week on 2/16:
midterm01-practice-lecture.pdf (compiled from lectures)
Online practice problems (compiled from homework)
midterm01-practice.pdf (solutions below)
02/12 lecture14.pdf Happy Mardi Gras!
02/14 review get ready for the midterm on Friday!
02/19 lecture15.pdf review midterm:
midterm01.pdf
midterm01-solution.pdf
02/21 lecture16.pdf Homework Set 5: Problems 1-2 (due 3/14)
02/23 lecture17.pdf Homework Set 5: Problem 3 (due 3/14)
02/26 lecture18.pdf Homework Set 5: Problem 4 (due 3/14)
02/28 lecture19.pdf new homework next class
03/02 lecture20.pdf Homework Set 5: Problem 5 (due 3/14)
03/05 lecture21.pdf Homework Set 5: Problems 6-7 (due 3/14)
03/07 lecture22.pdf new homework next class
03/09 lecture23.pdf Homework Set 6: Problems 1-2 (due 3/21)
03/12 lecture24.pdf Homework Set 6: Problems 3-6 (due 3/21)
03/14 lecture25.pdf Homework Set 6: Problem 7 (due 3/21)
03/16 lecture26.pdf start preparing for the midterm exam next week on 3/21:
Online practice problems (compiled from homework)
midterm02-practice.pdf (solutions below)
03/19 review get ready for the midterm on Wednesday!
03/23 lecture27.pdf review midterm:
midterm02.pdf
midterm02-solution.pdf
enjoy Spring break!
04/02 lecture28.pdf do Example 137; work through Example 144
04/04 lecture29.pdf Homework Set 7: Problems 1-3 (due 4/18)
04/06 lecture30.pdf Homework Set 7: Problem 4 (due 4/18)
04/09 lecture31.pdf Homework Set 7: Problems 5-6 (due 4/18)
04/11 lecture32.pdf Homework Set 8: Problems 1-2 (due 4/27)
04/13 lecture33.pdf work through Examples 162 and 165
04/16 lecture34.pdf Homework Set 8: Problems 3-4 (due 4/27)
work through Examples 169 and 171
04/18 lecture35.pdf Homework Set 8: Problem 5 (due 4/27)
work through Examples 173, 174 and finish 176
04/20 lecture36.pdf work through Examples 180 and 183
04/23 lecture37.pdf work through Example 185
04/25 lecture38.pdf start preparing for the final exam on 4/30:
final-practice.pdf (solutions below)
04/27 review get ready for the final exam!

## 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 lecture07.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 68 (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 111 (see lecture23.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

There will be two in-class midterm exams and a comprehensive final exam. Notes, books, calculators or computers are not allowed during any of the exams.

Our (tentative) exam schedule is:

The following material will help you prepare for the exams.